AFAIK - It’s not that there’s some special modern solution for dealing with the waste. It’s that the modern technology we have produces far less waste than one might think, because our perception of nuclear is outdated. There are also now ways to repurpose/recycle the waste which reduces total waste by 20-30%. I mentioned earlier in a reply to someone else that all of the nuclear waste created in the US since the 1950s can fit on one American size football field (stacked 10 yards high). That’s 70 years of nuclear waste that, properly stored, has a small geographical footprint and little to no impact on the surrounding environment.

I haven’t listened to these podcasts in a while, but if I remember correctly they do talk about modern methods for storing the waste that are much more efficient than in the past.