Back in the day when I was doing a lot of trap shooting, I liked to store the "target" shot shells in the 20 or 25 round commercial boxes they are sold in. I saved those boxes for hand loading those hulls.
A trap vest often has pockets on each side, one for empty hulls and another that a 25 round box will fit nicely into. This is very handy on the range.
There are also bags available with a pouch on the front for a 25 round commercial box and a large bag to contain the empty hulls.
I don't know how long you're planning on storing them, but many trap shooters, that reload, start reloading right after Christmas and continue to spring. They mostly store the reloads in those commercial boxes, which in turn are either stored on shelves or in cardboard boxes that contain ten boxes, which in turn are easily carried.
Long term storage is similar, as long as the storage area is dry and temps don't fluctuate a lot. Regular room temps are fine.