This is a complex issue, not just because storing radioactive material is complex, but because the "waste" are not a uniform single material. Some have a decaying process of 300 years (90% of the waste, actually), some have a much longer one.

In the beginning of the nuclear era, some wastes were... dumped in the ocean (it's as bad as it sounds). This is fortunately no longer the normal practice. Some dedicated storage sites are used to store them depending on their lifetime.

The latest solution is geologic storage (some caves were found with waste from naturally occurring fission, eons ago, radioactivity never escaped, so let's just... do that?). A site was identified in Finland with a hope it can store them for 100,000 years (of course, we don't have any reference that would last that long...). And the good thing is the storage is "reversible" for the first 100 years (if we change our mind/find better, we can still retrieve the waste during the first 100 years).

Finally, and that will resonate with @Waryle@lemmy.world comment: France had a 4th generation prototype reactor called SuperPhenix. Particularity of a 4th gen reactor is it can use some wastes to produce more energy. SuperPhenix being a prototype, it suffered from many issues through its lifetime. But at the end, it had a 90% uptime, and though it wasn't generating a lot of power (that was never the goal, remember: development...), some reports were recommending to keep it up so that it could have processed part of the existing nuclear waste.

To appeal to the ecologists party allied to the socialist Prime Minister at that time, SuperPhenix was definitely shutdown in 1997. And now, the same ecologists use the nuclear wastes issue as a big reason to push back any plan on nuclear power.

This is the entirety of the high-level nuclear waste that France produced for 80 years while having 70%+ of nuclear in its electricity mix.

The question of nuclear waste is an extremely minor problem compared to the ecological issues we're facing, and which we've been addressing for decades.

Anti-nuclear people just prefer to cover their ears and pretend it's an insolvable problem.

Most of it can be recycled (as in used for other nuclear products or services like MRI machines), but it doesn't because of fear of weaponization. What can't be recycled can be buried.

See these videos for more info on nuclear energy. The first one includes a nukeE's commentary. His intros are a bit dry, but he's very informative on kurzgesagt's content.