Earlier quoted context omitted.
Nuclear waste is not an actual problem and the only reason there is no 'solution' is because it has turned into a political shit show. Worrying about nuclear waste at the beginning of the nuclear age when most of that waste is actually valuable fuel is downright idiotic.
No, it actually is a problem. It has to be stored for millennia, and in a way that does not leak/contaminate surroundings (including ground water). Very little technology /materials can meet those requirements[0], and the current situation is a damn mess[1]. 0. http://www.nwtrb.gov/docs/default-source/reports/synopsis-of... 1. https://www.washingtonpost.com/news/post-nation/wp/2017/05/0...
Soon it will be possible to use most of the waste as fuel: "...Fast reactors can "burn" long lasting nuclear transuranic waste (TRU) waste components (actinides: reactor-grade plutonium and minor actinides), turning liabilities into assets. Another major waste component, fission products (FP), would stabilize at a lower level of radioactivity than the original natural uranium ore it was attained from in two to four centuries, rather than tens of thousands of years"
http://en.wikipedia.org/wiki/Integral_fast_reactor
https://en.wikipedia.org/wiki/Generation_IV_reactor
While there are issues with nuclear power, the worry people have about nuclear waste is greatly overblown to say the least. The amounts generated are manageable and in a relatively short amount of time we can use most of this "waste" to generate electricity. (The same can't be said for coal waste - like just about everything else associated with burning coal, coal waste is an environmental diaster.)