Much of the waste problem can be solved by using reactors designed to use the waste as fuel. This isn't currently done, but there's no big technical reason it couldn't be.
For what remains after that, well, there's nothing particularly wrong with sticking it in some extremely sturdy armored containers, sticking the containers in a shed, surrounding the shed with a razor-wire fence, and stationing a couple of guards at the gate.
The paranoia over nuclear waste is weird. What if civilization collapses and people in the far future discover the stuff and don't know it's dangerous? It lasts tens of thousands of years, after all! OK, but we leave dangerous stuff all over the place. There are tons of barrels of toxic chemicals, ponds full of poison, and heaps of awful stuff just sitting around, and more being produced all the time. Nuclear waste is bad because it lasts tens of thousands of years? How about arsenic or mercury, which lasts forever? And sure, precautions are taken with those toxins, but not on the "this must be placed in a geologically stable area, packaged to survive the fall of civilization, and signposted so that all future intelligence will know to avoid it" level of precaution.