Maybe this is a dumb question but couldn't we ship the waste to another planet (of course once we have rockets capable of doing so but that's not far imo).
We could fly it into the sun, the problem is that until we have a space elevator the only way we have of getting it out of the atmosphere is via rockets and a rocket explosion with a nuclear waste payload would be very bad.
Sure but since we're talking about pie in the sky stuff requiring tech we don't yet have to begin with, putting it into the sun is a better permanent solution.
I don't think Elon is ever going to colonize Mars, but other people may someday.
Potentially, but it is much, much safer to dispose of it here.
What's even better is to recycle it, because 95% of the original energy is still in the "waste". And when you do use all of it, the remainder remains radioactive for a much shorter period of time.