"Is it possible for humans to travel at the speed of light?"
Answer: No, not under our current understanding of physics. Which is incomplete and ever-expanding.
That's as good as you're going to get, for now. There's some approaches one way or the other, but more work needs to be done before the answer changes in a meaningful way.
And by the way, "Is it possible for humans to cross distances in a time shorter than it would take light to cross that distance in a vacuum?" is a very different question. It does have the same answer, for now, but it's the more likely of the two to actually change anytime soon. Not that it'll necessarily happen, just more likely.