I think you have a serious misconception. The long term storage we're talking about here is for the spent fuel rods (or alternatively the byproducts extracted from them if they are reprocessed). It's known as high level waste and there isn't very much of it.
https://en.wikipedia.org/wiki/Radioactive_waste
> As a general rule, short-lived waste (mainly non-fuel materials from reactors) is buried in shallow repositories, while long-lived waste (from fuel and fuel reprocessing) is deposited in geological repository.
> Overall, the 60-year-long nuclear program in the UK up until 2019 produced 2150 m3 of HLW.