The challenge with shooting nuclear waste out into space is that you have even greater containment issues.
Even if you decided to rocket it all into our sun or into Beetlejuice, the cargo could be destroyed by meteor showers or knocked off course to crash land on asteroids or other planets. We may one day mine asteroids for resources: I don’t want our grandchildren to be SOL because an asteroid had Chernobyl level radioactivity.
Thanks so much for the Quark show reference, I never knew about it; the short clip had me laughing out loud!
I think sending that stuff in space, far away towards a back hole, or even towards the sun, once on its way at a safe distance from Earth, would pose an infinitesimal problem, if any at all, for anyone in the future, given the probabilistic realities. And it may prove actually efficient cost wise vs. the storage example in @yaypie OP, which to me today looks more like due to some reason or another it turned an once active power producing plant, into toxic residue storage facility acting as milk cow for cash in disguise, multiple generations will have to contend with. But I am no scientist and feel very short sighted regarding society's future..