Here's an inexpensive storage idea: Visit your area thrift shops and look for one of those all-cedar wood upright storage closets -- the kind from the early to mid 1900s. I even made one of these in wood shop class back in 1965. It is like having one huge upright Cedar chest, plus you can hang items in it, too. The last one I saw had a $45 price tag on it. If you do find one and it does not have that familiar Cedar aroma, do not worry. All you need to do is give a light sanding to the wood inside and the aroma is back again. Just be sure to clean out the dust afterward. A vacuum cleaner followed by a tack cloth would do just fine.
Not only will you have something that will be a great storage device, but it will cost you a fraction of what is available for storage today. Oh yes, I would think it wise to place a non-corrosive rack in the bottom, layered on top with archival paper, to keep the bottom quilt from making direct contact with the wood.