I store my fabric in cardboard because I don't have any issues with bugs and it keeps the light off the fabric, I've had some of my fabric for decades and light degradation is an issue.
Can't always afford acid free boxes, but acid free tissue paper is relatively cheap, easy to get and store, and lines those boxes nicely.
I do have archival quality tags to to use for notes and brass safety pins. For quite awhile I collected vintage fabrics intending to sell them on Ebay and used those for any comments along with dimensions. Never did get going on that and sold my boxes of yardage to someone who does and who goes to quilt shows.
But again, some of them had been stored for 20 years that I had them with no issues.