The primary issues with long term management of nuclear waste are political and cultural, not technical.
A key thing here is that there are two schools of thought on how to manage waste:
One idea is to reprocess it to separate out the hottest materials, which are then reused as fuel. The hot stuff then never needs to be disposed of, only low-level waste. That seems to be favored by most nuclear advocates but is politically unacceptable due to concerns about proliferation and the need to transport materials to and from reprocessing facilities.
The second idea is vitrification and deep geological disposal - pulverized everything, mix with molten glass, let it cool and stick it deep underground. This could happen with or without reprocessing, but is usually advocated alongside no-reprocessing. Then it gets controversial because they are burying stuff that is still very radioactive and will be for tens of thousands of years. (There is also the separate issue of misinformation suggesting that very long half-life = really dangerous. Once you get into half-lives of millions of years, you are dealing with stuff that is not much more meaningfully radioactive than a granite countertop).
I'm all about reprocessing and reuse as fuel. The most dangerous of the dangerous stuff never needs to be in the waste stream as it currently is.