Mine are stored flat in the original envelopes they came it. I have cats and forced hot air heat. Getting cat hair and dust off the dies can be a pain. I did card crafting with the Sizzix for years and had the same problem with cat hair and decided I was going to keep the Go dies covered. I had an old cabinet that just fit the Go.
The Go itself stays open, but covered with a towel. The Siamese perches on it.