Most cardboard isn’t archival. I have a Master’s in Library Science with an emphasis on Special Collections Preservation, and so I have the background to give you my informed opinion: I don’t give a fig about archival storage. Seriously, friends, it’s all going to end up dust anyway someday, and archaeologists 3,000 years from now don’t need to see my fabric stash OR my wedding scrapbook, thank you very much. If you want an archivally-safe cardboard, however, then know that it does exist; just check out any web site or brick-and-mortar store that caters to libraries.
I folded very large pieces and thick pieces of fabric, such as felt and fleece and faux fur, around some larger broken-down shipping boxes that I own. It’s not plastic, but it’s no longer recycling, either to do that, since I hadn’t actually used the shipping boxes yet–search for sources of used boxes to break down yourself, to keep a recycling emphasis on this project.
I still throw fabric that’s too small to fold around cardboard into bins, sorted by how I use it–novelty quilting cotton, patterned quilting cotton, novelty and patterned flannel and fleece, felt, and everything else. When a bin gets full, perhaps less than once a year, I sort through it and put together a bag of scraps to take to our Recycling Center’s re-use shop, where it’s donated to local arts programs and non-profits. I have a dedicated space for my folded fabric to live, as well, no cramming allowed, and when that space is full…well, the local arts programs get another donation!
Last thing–when I first transitioned my stash to this system, yes, it took HOURS to do. Days, even. Now, however, when fabric comes into my house, I wash it, dry it, and then spend less than five minutes folding it around cardboard and setting it in its correct spot on the shelf. Finally, simple storage!