Sometimes it helps protect the item if you store it in it's original packaging. I have a few Impressabilities metal embossing plates from Spellbinders; they are delicate and I like to store them in the cardboard sleeve they came in. Although, the newest one did not come in the protective heavier cardboard so I had to make one from something else.
If you are not storing anything in them; and you just have a bunch of empty packaging taking up space; you have to decide if you are the kind of crafter who will "up cycle" any of it or not. I've seen people cut up that heavier plastic and emboss it and use it on 3-D projects. While I like to see what others have done; I know I am not that kind of crafter. I save cardboard that can be cut down to envelope size to help protect any "lumpy" cards I might make in the mail; and I have a few empty clamshell containers from SU sets that have been repurposed as storage for other small embellishments and that's it. Anything else that can be recycled goes into my recycle bin.
Mary in NC