Yes, I ask you also, true companion, help these women, who have labored side by side with me in the gospel together with Clement and the rest of my fellow workers, whose names are in the book of life.

Clement, through his writings and the writings of others, came into contact with Peter, Paul, and possibly John. Clement, not an eye witness to Christ, was an eye witness to these other men.
Irenaeus claimed Polycarp and Ignatius studied under the Apostle John. Polycarp and Ignatius made the same claim, treating John as an eye witness to Christ. Clement, an eye witness to Peter and Paul for sure, documented their existence and their claims to be eye witnesses to Christ.
Peter, John, Matthew, James and Jude all wrote books of the Bible claiming to be eye witnesses to both Jesus and the events of his life. Then there is Paul, who we know persecuted the early church, then claimed a supernatural physical visit from Christ after his death. The other church leaders who he had sought to kill took him into the church and affirmed his ministry. But we do not even have to get to Paul to establish this — either Jesus existed or a great many people over a century collaborated in an elaborate conspiracy to create him.
To claim Jesus did not exist, we must also declare a bunch of other people — who we know existed by their own writings and the writings of others — did not exist.
So that all leads to the next question:

If Jesus existed, why did so many claim him to be God?

Here, I have to give a good bit of credit to Pastor Mark Driscoll and his sermon on James. Driscoll is getting a lot of criticism these days over plagiarism allegations. I am reading the book in question and will address that at some point. But for now, just know that Driscoll’s sermon is Biblically based and Biblically sound. Also, I do like Driscoll, would very much like to meet him, and think he is worth reading. I’ll add reservations and caveats about his book at a later date. Suffice it to say, I do not think the controversy disqualifying.
So, to get to Jesus’s claims about himself and others’ claims about him, we first need to broach an issue. The Bible claims he had brothers. At the Council of Constantinople in 553 AD, the early church declared that Mary was “ever virgin.” Many Christians believe this. It is not just a Catholic belief. Early Protestant leaders like Luther, Calvin, Wesley, Zwingli, and others agreed. They interpreted the references to Jesus’s “brothers and sisters” as either (1) Joseph’s children from a prior marriage or (2) his closest cousins in an extended family.
Going into this, understand I think Jesus’s brothers and sisters were his half-brothers and sisters, all younger than him, from the marriage between Mary and Joseph. But for purposes here, we should all agree that, at least, his “brothers and sisters” were his closest family who knew him best — whether half siblings or closest cousins. There are a number of passages that reference them in the New Testament and the sense of the phrasing is that they were his closest relatives.
Mark 6:1-6 describes the family thusly:

He went away from there and came to his hometown, and his disciples followed him. And on the Sabbath he began to teach in the synagogue, and many who heard him were astonished, saying, ‘Where did this man get these things? What is the wisdom given to him? How are such mighty works done by his hands? Is not this the carpenter, the son of Mary and brother of James and Joses and Judas and Simon? And are not his sisters here with us?’

Matthew 13:53–57, in accordance, reads:

And when Jesus had finished these parables, he went away from there, and coming to his hometown he taught them in their synagogue, so that they were astonished, and said, ‘Where did this man get this wisdom and these mighty works? Is not this the carpenter’s son? Is not his mother called Mary? And are not his brothers James and Joseph and Simon and Judas? And are not all his sisters with us?”

Jesus had four brothers: James, Joseph, Simon, and Judas (later called Jude). He had at least two sisters. The tradition at the time was the oldest son typically received the grandfather’s name. We know that Joseph’s father’s name was Jacob. Matthew 1:16 tells us, “and Jacob the father of Joseph the husband of Mary, of whom Jesus was born, who is called Christ.”
James is the Greek derivative for Jacob. We can conclude that James was either Joseph’s oldest natural born son or the oldest son of Joseph’s own brother. If the second son was indeed Joseph’s son, it makes sense the first son is named for the grandfather and the second son for the father himself.
This also explains why there are so many Jacobs, James, and Judases in the Bible. Jacob, in particular, was very popular given Genesis.
Many people may not realize that, based on the eye witness accounts of Jesus’s friends, Jesus’s family thought he was a nutter. So much for the “Liar, Lunatic, or Lord” framing. His family was all in for lunatic. See Mark 3:21, 31–35:

“And when his family heard it, they went out to seize him, for they were saying, ‘He is out of his mind.’”
“. . . And his mother and his brothers came, and standing outside they sent to him and called him. And a crowd was sitting around him, and they said to him, ‘Your mother and your brothers are outside, seeking you.’ And he answered them, ‘Who are my mother and my brothers?’ And looking about at those who sat around him, he said, ‘Here are my mother and my brothers! For whoever does the will of God, he is my brother and sister and mother.’”

Today, as Mark Driscoll and others have noted, we would call this an intervention. Jesus’s “mother and his brothers came” trying “to seize him” because they thought he was a nutter claiming to be God. The most extraordinary thing about this is that Jesus’s own mother was involved. Luke 1 tells us the Angel Gabriel appeared to Mary and told her “He will be great and will be called the Son of the Most High.”
Mary clearly knew he was special and from the Lord. John 2:1-5 — an eye witness account — tells us

On the third day there was a wedding at Cana in Galilee, and the mother of Jesus was there. 2 Jesus also was invited to the wedding with his disciples. 3 When the wine ran out, the mother of Jesus said to him, “They have no wine.” 4 And Jesus said to her, “Woman, what does this have to do with me? My hour has not yet come.” 5 His mother said to the servants, “Do whatever he tells you.”

Despite what she had experienced and knew, Mary too went with Jesus’s brothers to seize him and carry him home. Note that Mary stuck with Jesus the whole way through his life, unlike his brothers and sisters — no doubt coming to a richer and richer understanding of her son over time.
John tells us Jesus’s brothers wanted him gone. John 7:2-5 notes

Now the Jews’ Feast of Booths was at hand. So his brothers said to him, ‘Leave here and go to Judea, that your disciples also may see the works you are doing. For no one works in secret if he seeks to be known openly. If you do these things, show yourself to the world.’ For not even his brothers believed in him.

This is Jesus’s brothers confronting him telling him that if he really thinks he is a big deal — if he really thinks he is God — he needs to go to the big city and show everyone. He needs to tell the world, which he can’t do in a small town. They want him gone, with his friends, and given the implications of what they’re telling him to do, they may very well think he is going to get himself killed.
The brothers who had tried to stage an intervention had given up and wanted their brother gone. And Jesus goes. He winds up being arrested, tried, tortured, and crucified. The most striking thing here is that his brothers did not even show up at the execution. His mother was there. The mother, who with the brothers, had tried to save Jesus from himself — she was there. But the brothers were not.
From the account in Matthew 27:55,56

There were also many women there, looking on from a distance, who had followed Jesus from Galilee, ministering to him, among whom were Mary Magdalene and Mary the mother of James and Joseph and the mother of the sons of Zebedee.

From Mark 15:40,41:

There were also women looking on from a distance, among whom were Mary Magdalene, and Mary the mother of James the younger and of Joses, and Salome. When he was in Galilee, they followed him and ministered to him, and there were also many other women who came up with him to Jerusalem.

From Luke 23:49:

And all his acquaintances and the women who had followed him from Galilee stood at a distance watching these things.

Lastly, from John 19:25-27:

but standing by the cross of Jesus were his mother and his mother’s sister, Mary the wife of Clopas, and Mary Magdalene. When Jesus saw his mother and the disciple whom he loved standing nearby, he said to his mother, “Woman, behold, your son!” Then he said to the disciple, “Behold, your mother!” And from that hour the disciple took her to his own home.

From John we learn that Jesus, from the cross and about to die, told John that he had to look after Mary. We learn that “from that hour the disciple took her to his own home.” Why? The brothers who had tried to stage an intervention would not show up at the execution. Mary was there with no immediate family. John, the Apostle, had to take her into “his own home.”
And Jesus died.
If that were all there were, lunatic he would be. The family would have been right. They tried to intervene to no avail. The brothers sent Jesus packing. He got himself arrested, tried, and killed. They wouldn’t even show up as he hung on the cross dying. Or at least we have a record of who was there and not one of those eye witness accounts documents his brothers being there. Jesus’s best friend is commanded to take care of Jesus’s mother as if they were son and mother.
That would be the end of it, except something extraordinary happened.
From Acts 1:14 we learn that “All these with one accord were devoting themselves to prayer, together with the women and Mary the mother of Jesus, and his brothers.”
It is clear from the text that these are not the Apostles. These are James, Joseph, Simon, and Jude — these are the family members who had tried to seize him, urged him to leave, and would not show up at his death. They were in the early church. So what happened? Seriously? These people thought he was crazy. They, his family, knew him best. Were he some sinner or a jerk they would not make up the early church after he, the lunatic jerk, had died. But there they were.
Look at James alone. James became a leader in the early church. Paul called him a pillar. Paul traveled to Jerusalem after his conversion to meet with the Apostles and with James. This is James the brother of Jesus, not James the Apostle. James the brother of Jesus, called James the Just, came to be referred to as “camel knees” because he was on his knees praying so much.
James the brother who had rejected Christ in life became a pillar of the early church vested with authority.
Paul, writing to the Galatians documents

“When they saw that I had been entrusted with the gospel to the uncircumcised, just as Peter had been entrusted with the gospel to the circumcised (for he who worked through Peter for his apostolic ministry to the circumcised worked also through me for mine to the Gentiles), and when James and Cephas and John, who seemed to be pillars, perceived the grace that was given to me, they gave the right hand of fellowship to Barnabas and me, that we should go to the Gentiles.” (Galatians 2:7–9)

From Acts 15:12-21:

And all the assembly fell silent, and they listened to Barnabas and Paul as they related what signs and wonders God had done through them among the Gentiles. After they finished speaking, James replied, ‘Brothers, listen to me. Simeon has related how God first visited the Gentiles, to take from them a people for his name. And with this the words of the prophets agree, just as it is written,
“After this I will return, and I will rebuild the tent of David that has fallen; I will rebuild its ruins, and I will restore it, that the remnant of mankind may seek the Lord, and all the Gentiles who are called by my name, says the Lord, who makes these things known from of old.”
Therefore my judgment is that we should not trouble those of the Gentiles who turn to God, but should write to them to abstain from the things polluted by idols, and from sexual immorality, and from what has been strangled, and from blood. For from ancient generations Moses has had in every city those who proclaim him, for he is read every Sabbath in the synagogues.’

Paul sought out James and Peter — Paul, the guy Jesus himself had told to go preach to the Gentiles, went to find Jesus’s brother who had rejected Jesus in life. He didn’t just seek him out, we learn from Paul and others that James had, in fact, become a well known and established authority figure in the early church. He considered himself a servant of Jesus, not his brother — a servant to a living God who had been crucified.
In 62 AD, early church history notes that the local Jews of Jerusalem went to James. They respected him. They told him they wanted him to tell all of Jesus’s followers that, being Jesus’s brother, he could testify Jesus was not God. Made sense, didn’t it? Here’s the guy who escorted Jesus out of town and wouldn’t show up to the funeral because his brother was an embarrassing nutter. Also, here is a guy, being Jesus’s brother, who could claim part of Jesus’s legacy and become the icon himself.
But by 62 AD, James was so invested in the idea that Jesus was the Risen Lord he told the Jews the crowd was right. Jesus was Lord. Enraged, the Jews carried him to the top of the temple and threw him off. When he did not die, they stoned him and beat him with clubs until he died.
Then, the early church tells us, Jesus’s brother Simon took James’s place.
Along the way, Jesus’s brother Jude also became a church leader. He too eventually was killed by the Roman state in a purge of Christians. Accounts are mixed as to whether it was his children or grandchildren, but it appears his grandchildren were called before the Emperor. They testified that that their relative Jesus had been talking about a return at the last day, not an imminent take over of the Empire — that he was King in Heaven. They were spared, became leaders within the church themselves, and were executed by a later Emperor.
Jesus’s family, who had rejected him in life, were willing to die proclaiming he had risen. Something had to have happened. Paul wrote to the Corinthians that Jesus “appeared to Cephas, then to the twelve. Then he appeared to more than five hundred brothers at one time, most of whom are still alive, though some have fallen asleep. Then he appeared to James, then to all the apostles.” (1 Corinthians 15:5-7)
It is a silly thing to say that Jesus did not exist. There is an ample historic record to show, through eye witnesses, that Jesus and Socrates both existed. Many atheists concede Jesus existed, but, unlike with Socrates, they say extraordinary claims require extraordinary evidence.
One of Christ’s friends betrayed him, then committed suicide.
Eleven of the twelve who followed Jesus were willing to go to the ends of the earth to proclaim him God.
Ten of the twelve met terrible deaths because they would not recant that Jesus was God. The tenth, John, lived in exile. His students and others documented the numerous attempts to kill John.
Jesus’s brothers, who rejected him in life, embraced him as a risen, living God after his death. They too were willing to be put to death for refusing to recant after Christ’s crucifixion what they refused to believe when he walked and talked with them.
Others came claiming to be the messiah. Their claims did not last. The man named Jesus not only must have been a spectacularly charismatic person, who surrounded himself with spectacularly charismatic people — all of whom were willing to be tortured and killed — because Jesus and these men were able to recruit into faith a lot of others who, over two thousand years, grew into the world’s largest religion. Many of them were persecuted, tortured, and killed in horribly gruesome ways. Still they persisted in the faith.
So either these men were charismatic liars so invested in their lies they were willing to be tortured and killed or they were telling the truth.
Those who do not want to believe will not believe. As for the rest of us — Christ’s own family rejected him as a lunatic then, after his torture and crucifixion, picked up the cross claiming Christ had risen. And they too, the brothers who rejected him in life, were willing to die proclaiming him risen.
That’s pretty extraordinary to me.