To store your quilts, avoid direct contact with cardboard boxes, wooden drawers, or wooden shelves. Cardboard and wood are acidic and will deteriorate and yellow fabrics over time. You can find archival boxes for quilt storage as well as acid free tissue for lining boxes and shelves.
Plastic containers can be used for quilt storage if they are made of new High Density Polyethylene (HDPE) or Polypropylene (PP). New plastic containers are typically clear or milky, not colored, and look for the #2 or #5 symbol on the bottom. Be sure to line these plastic containers with acid free tissue or a clean, unbleached cotton sheet. And be sure that there is no moisture present in plastic containers as it can lead to mold and mildew growth.
If you have a spare room, take advantage and store your quilts spread flat on a spare bed with clean, unbleached cotton sheets in between. If you fold your quilts, be sure to refold them every couple years to avoid creases.