I don't have a huge collection of dies (about 25 in total) so I store them in a Sterilite bin with locking sides. I can pick up the bin and move it off my shelf when I want to use them and just lock the top and slide it back in when I'm done. The dies can stand up on end in the bin, which is what is recommended (so the foam doesn't get compressed).
It's similar to these:
https://www.walmart.com/ip/Sterilite...of-10/44785807