That's a tough one, but as mentioned above I would definitely avoid cardboard. Ideally fabrics are able to "breathe" but it sounds like the best bet is plastic totes. Sterilite ones should be archival safe:
http://www.sterilite.com/general_info.html,
http://takecareofyourstuff.blogspot....ival-from.html, nd this link in particular has some good info about storage and general preparation:
http://www.preservesmart.com/baby_cl...eservation.htm
I use Sterilite drawers from Target to store my vintage and antique fabrics. I have stayed with white and clear colors though in case because to me the less extra ingredients (i.e. dyes) the better.