For years I have been storing my stash in plastic underbed boxes. I recently outgrew the two I had been using, and they were in sad shape: cracked and broken along the edges where the handles were from pulling out. Soooooo heavy! So I've upgraded to underbed boxes on wheels. These are higher than what I had, so the bed had to go up on risers. We're using 8" risers, so there may be enough room for stacking! (tee-hee-hee)
I have metal closet doors, so I can just use magnets to put things up on there when I want to try them out. I have also used wool or felt held up with the magnets and then I can gently press the individual pieces in place and voila.
However, as for the fading problem, let me just say this: there are some fabrics that may fade by being in the plastic boxes, too. I have three fabrics bought for me many years ago by DH for a specific project (as yet unstarted, of course!). When I was transferring the fabrics into their new home, I noticed that they each have a little bit of facing along the top folded areas. Not the bottom folds, just the tops ....
Just for your info, but it may help.
Odessa