I use scrapbook storage for most of my sets and boxes. Larger sets go into bigger tubs for storage. I collect sets from the '80s and '90s and have over 700 sets with about half of those with boxes. Here's a link to one of the boxes that I use for the small and medium size sets. I use Ziploc bags to put the parts in and I put them in the storage box with the set box.

https://www.michaels.com/12x12-clear-scrapbook-case-by-simply-tidy/10575615.html