I have been using cardboard for my stash for some 20-40 years. I've never had fabric discolor from the cardboard. The big think to watch for is moisture wicking up through the cardboard -- cardboard boxes should
never be stored directly on a concrete floor (like basement or garage). Trace amounts of moisture are the typical cause for the musty smell and also attract bugs.
If you are concerned, the easiest solution (which I did) is to buy some archival tissue paper and line the boxes. If it is going to be a regular storage solution, you can even buy archival boxes.
As for size, I like the sturdy banker boxes I get from Costco or office supply stores designed for document storage. It is worth a few extra bucks for the heavy duty, the light ones do not hold up well. They are a great size for fabric, even full can be handled by the average female, they stack nicely up to 4-5 high. Personally, I like having separate lids that interchange instead of flaps. I write the contents on the outside of the box.
https://www.costco.com/bankers-box-h...100355478.html