The original packaging is probably the way to go for you, especially when your storage situation isn't permanent. Clear bags, like the ones many brands are already packaged in, are cheap and easy to find for those sets that don't already have them (like Papertrey).
I started stamping when I was in college (10+ years ago) and kept everything in one of these
Sterilite 3-drawer units. It wasn't ultra-portable for crafting away from my desk, but it was small, easy to use, and a cinch to move (just secure the drawers with packing tape and you're good to go). I kept paper in one drawer, stamps in another, and inks/tools in the last. When I had room to spread out in my first apartment, I bought another. And later, another. I still use the originals plus half a dozen more. They've been stacked in a tower next to my desk, lined up on a desk to create a short hutch, and filling wall-to-wall shelving depending on my living/crafting space. They're my go-to storage pieces and I use them every single time I craft.