99% of nuclear waste, and basically all the long-term stuff, is transuranics. Those are fuel, for fast reactors like this one. Consequently they produce 1% as much waste as conventional reactors, and it goes back to the radioactivity of uranium ore in about 300 years.

Another consequence is that these reactors can use most of our existing stockpile of nuclear waste as fuel. It's the only technology that can actually reduce our waste stockpile instead of just storing it somewhere.

Storage for 300 years is generally considered a solved problem. I think the basic idea is to encase the waste in blocks of glass and bury them somewhere.