It's called "closed [nuclear] fuel cycle". Just google it. I studied it at a university.
TLDR; if you have enough fast neutrons, you can transmute anything into safe materials. Fast neutron reactors produce enough, classic PWR reactors do not. The only commercial fast reactor right now is in Russia.
If at some point humanity decides to stop making reactors altogether, it's still possible to burn the waste with particle accelerators. It'll take hundreds of years, but waste won't be going anywhere.
And finally, if commercial fusion reactors ever happen, they can also be used as neutron sources to trivially burn up all the waste.
There's very little waste that lasts hundreds of years, and the reason it's "prohibitively expensive to store" is purely political. Because we safely and cheaply store it now while waiting for multi-decade trillion-dollar projects drilling deep mountain storage close to magma or something.
See page 15: https://international.andra.fr/sites/international/files/202... Only 0.2% of all waste is High Level Waste that is both long lived and highly radioactive.
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In the US reprocessing of civilian nuclear waste was stopped not for technical reasons, but for political reasons. The primary reasoning was that: US reprocessing of civilian nuclear waste would encourage other non-nuclear weapon states to build nuclear reprocessing capabilities which would make easier access to plutonium - nuclear weapon material.
"On April 7, 1977, President Jimmy Carter announced that the United States would defer indefinitely the reprocessing of spent nuclear reactor fuel. He stated that after extensive examination of the issues, he had reached the conclusion that this action was necessary to reduce the serious threat of nuclear weapons proliferation, and that by setting this example, the U. S. would encourage other nations to follow its lead."
https://www.pbs.org/wgbh/pages/frontline/shows/reaction/read...
Commercial fusion reactors could be used burn (transmutate) long-term transuranic waste, on the other hand they will produce short-term nuclear waste, like neutron activated steels.