The reason I made the large case for storing thread, is because I am concerned about the sunlight causing the cotton thread, which I use, to degrade over the course of time. I would guess if you had a storage rack that was on the back of the closet door, , and the door was kept closed, for the majority of the time, it would be fine. It would be very easy to make a rack out of 2" x2"
lumber and then holes drilled and spaced appropriately, and dowels inserted to it to hold the spools. The whole thing could be made and sized to your door and look great, when all those various colors were arranged in a graduated pattern. That would be my choice, for a home made type of storage rack.
John