You do not want to store in plastic bags because of outgassing from the plastic. This can have deteriorating effects on fabric.
Hard plastic storage bins do not outgas, but you have the problem of trapping moisture inside the bin. The trapped moisture will be bad for the fabric. That's why wedding gowns are preserved in vacuum boxes; all the air is removed to prevent moisture damage to the fabric. Plastic bins still might be an option for you if you drill holes in the plastic to allow for air circulation and/or incorporate packets of moisture-eaters (sorry, I forgot the name -- it's the same stuff that is in the little packets that come inside bottles of vitamins, and is sold in stores for flower preservation).
As others have mentioned, quilts need to be re-folded every few months if they are stored folded. Folding on the diagonal (corners to center first) is best because it distributes stress along the bias (which means many threads bear a little stress instead of a couple of threads bearing all of the stressof the fold).
The best solution is to store flat on an unused bed, covered with a sheet. Quite a few quilts can be stored on one bed this way. Ideally the bed should be big enough so the quilt doesn't have to hang over the edge. Benefits to storing flat are that there is no stress along fold lines. The quilt does need to be covered to protect not only against dust, but against the damage that can be done by light. Not only sunlight, but also room lighting work to deteriorate fabric.