As Prism99 said, storing in bare cardboard isn't great longterm. My fabric is stored in 2 ways: 1) in fabric-lined baskets, further wrapped in white feedsack dishtowels and 2) banker boxes lined with freezer paper (waxy side out) and wrapped in groups with white feedsack towels or muslin. For fabric liners, I'd get a JA coupon for 40% or whatever off one fabric cut and buy yardage for wrapping. Or use old sheets.
I found some very cool white baskets at WM online, lined with white fabric (and no smell!!!) to use for scraps sorted by color.
hugs,
Charlotte