Here's a mindscrew for you. We see Clara on Gallifrey, but presumably she is still biologically human. If she went to every point in his timeline, she must have been there right at the beginning of his very existence.
What if one of the things she did to ensure his safety was to create him in the first place, Emissary-style? What if, not knowing the significance of her actions, she shagged a Gallifreyan bloke and gave birth to a baby?
That would make the Doctor half-human, on his mother's side.