Some years ago I explored this question in some depth. At the time, I thought it was 50-50 ─ in particular there was no need for an HJ in order to account either for the cult or the gospels (&c), none of the authors ever met an HJ, contemporary historical records are totally silent, and so on.
These days I'm more inclined to think there was an HJ than that there wasn't. I was influenced in that direction by Ehrman's point that none of the early opponents of Christianity every used the argument that the hero of the stories never existed; and by the 'criterion of embarrassment' quality of the portrayal in all four gospels of Jesus fighting with and snarling at his mother, and in places his family (Mark 3:31-35, Mark 6:4-5, Matthew 10:35-37, Luke 11:27. John 2:3, contrast John 19:26).
But Mark, the template of the other three gospels, has the earmarks of a composed story by someone with a Greek-influenced education (though, so I've read, he wrote the roughest Greek of the four), and so gives his tale a beginning, a development, and a climax, is familiar with the Tanakh and continually exploits it for his hero's benefit, and is unambiguously tendentious in selling its hero. It certainly implies support for Paul's claim that there was a Jesus cult in Galilee and Judea at the time.
So if there was indeed an HJ, as I suspect but do not strenuously assert, I seriously doubt we'll ever know much about him. It seens reasonable to guess that he was born, grew up, and first came to notice in the sticks, that he was a player in the Jewish religion industry, that he was a supporter of the apocalyptic strain of Judaism flourishing at the time, (and all three synoptics have him promising that the Kingdom would be established on earth within the lifetime of some of his audience) and that he was likely crucified by the Romans, and if so for being a political stirrer.