If done properly, all that needs to be stored are the fission products, with the recovered fissile material put back into the fuel cycle. The fission products are mechanically and chemically stabilized via a vitrification process. This is how the French have been doing it for decades, it is one thing that the French has got right.
France has more of their power from nuclear than anyone, but by reprocessing it they have very little waste to store. They don’t need any Yucca Mountain. Our peanut brained alleged nuclear engineer, the worst Annapolis graduate ever, still prevents us from doing the same via his executive order.