Prism99

I would simply place a quilt inside a large pillowcase (can make one to size out of an inexpensive, washed flat sheet) and place that inside a clear storage bin. Plan on re-folding the quilt every year or so. I would pad folds with left-over batting if you have it on hand; cheaper than acid free tissue paper.

It is not safe to put fabric into direct contact with wood products of any kind, including cedar. If you had a new cedar chest made, you would have to make sure the interior is finished with a clear poly coating or something similar that would prevent wood oils and acids from seeping into your fabric. Really, the cedar in a cedar chest is most useful for keeping moths and mice out -- not so much for preserving fabric. The wood in very old cedar chests has dried out enough over time that it is less likely for oils and acids to leak out of the wood. New wood is a completely different story.

Hard plastic does not outgas a lot, so I would not be concerned about it -- especially if airing the quilt out every year. An advantage of a hard plastic bin (with securing handles) is that mice cannot chew through it (as they can with a cardboard box, archival or not) and the bin will float in a flood.

Edit: I would do the same with the t-shirts. For them I would probably place a sheet of archival tissue paper between each shirt just to make sure none of the inks or dyes "travel" to another shirt. Oh, and place a few packets of dessicant in the storage containers; the last thing you want to do is seal moisture into a plastic bin along with your t-shirts and quilts. Long-standing moisture is terribly damaging to textiles.

One more thing, obviously, is to store in a dark area. Aside from moisture, light of any kind can fade cotton fabric faster than you might think.