To the more or less the same extent that most people in history have been confirmed to have existed, yes.
Contemporary records in the ancient world are rare. Beyond Roman emperors and governors, they are vanishingly rare; for example, there are zero surviving contemporary references to Hannibal, despite the profound importance of the Punic Wars on the evolution of Rome. Jesus was a Galilean peasant preacher; it is utterly expected that there would be no surviving references to him until decades after his death.
There are many references to Jesus written soon after his death; many of these were later collected into the New Testament of the Bible. Mark was the earliest gospel to have been written, and it is believed to have been written about CE 66-70. It draws heavily from the apocryphal Gospel of Thomas, the earliest parts of which were written between 40 and 60 CE--within recent memory of Jesus' death, which is likely to have occurred approximately 33 CE. The Epistle of James was possibly written 50-60 CE, and most of the Pauline epistles were almost certainly written 50-60 CE. To speak roughly, the Crucifixion was about distant in time from the earliest surviving Christian writing as 9/11 is from the current date, and about as distant from the first Gospels as the death of Princess Diana is from the current date. While we obviously cannot unquestioningly accept the entirety of the Epistles and Gospels as fact (nor can any primary source ever be unquestioningly accepted), few mainstream historians seriously consider that the entire thing is likely to have been made up out of whole cloth.
First, it doesn't really fit contemporary expectations of the Messiah. The Messiah was supposed to be of the house of David, from Bethlehem, who would lead the Jews to eternal glory and establish Israel as the greatest power on Earth, vanquishing all enemies of God beneath the sword. Jesus was a homeless man of unknown lineage, son of a rural laborer, from Nazareth, and as soon as he arrived in Jerusalem he was executed in the style of slaves, thieves, and brigands. If Jesus were made up, why make him from Nazareth? Why make his story so counter to the audience's expeectations--why not make him a heroic military rebel? And why have him be crucified, a shameful and degrading punishment that was absolutely unthinkable for a holy man to endure, and preposterous for a god to endure? It would have been (and, according to the later gospels, actually was) a completely unbelievable story to contemporary Jews. For a comparison, perhaps, a modern analogue might be if I were to claim that Jesus indeed returned, but actually he was a dude named Mike from Detroit this time (but, uh, he was born in Bethlehem because his parents were on vacation there), and Mike got shot by the cops while robbing a liquor store instead of creating the Kingdom of Heaven on Earth. If I were to try to start a religion, and make up my Messiah out of whole cloth, I would probably not make the Messiah a small-time crook from Detroit. Similarly, if given a choice, the early Christians would probably not have made the Messiah a crucified itinerant from Nazareth.
Second, the earliest gospels and epistles were written at a time when Jesus was within living memory. If making up a memorable, charismatic figure, why say that he lived where your primary audience lived, and why say that he lived and died within living memory? If making up the man, it is unlikely they would have pointlessly run the risk of the audience saying, "Hey, wait, I remember dozens of preachers and holy men, and I don't ever remember anything about Jesus."
Third, there are near-contemporary references to Jesus outside the Christian community, in Josephus and Tacitus. This actually makes Jesus somewhat better attested than we might expect, considering how slight the evidence is for most historical figures of the time.
Ultimately, the historical Jesus is boring. He was a small-town preacher who wandered around, got a few followers, and then went to the big city and got himself in trouble with the cops, and then got himself killed. From scattered contemporary records, we know of at least a dozen similar historical stories. While he must have been exceptionally charismatic, in broad strokes his life story was so utterly unremarkable that it wouldn't even be worth falsifying.
