If two contestants are that close they can both have a gold medal. Ties are a perfectly acceptable outcome.

That is also fair. I think both are fair enough in this rare situation. Whether or not the finish is this close is already luck-based even when two people have equal skill.

Or have them race again (and again, and again...) until one comes out ahead, like how ties are broken in other sports too.

Why randomly pick? Give them both gold medals.