I came up with a five-drawer blueprint flat file. These handy-dandy files are made for (duh) blueprints, but hold fabrics beautifully. I can lay a fat quarter out nice and flat in a drawer. The flat files come in several sizes and also in a ten-drawer model. Mine is the smallest at 40"w x 28"d x 16" h. It is made of metal like any file cabinet. You can order a stand for it, but DH is going to incorporate mine into a custom-built wall cabinet. Eventually.
You can find these at larger high-end office supply stores. I ended up getting mine online (Global Industrial) as it was cheaper for me to do it this way (even though shipping was over $100.00) than order it in through a local store. They can be found used, but I didn't have much luck with it as I don't live in a large metropolis.