My upline has a lovely stamping area, and she stores all the SU colors by color. Paper, designer paper, ribbon, inks, refills, markers, embellishments...the whole nine yards. And it is all in drawers by color.
This picture would be overwhelming to anyone. But there's 210 drawers there, and you wouldn't need anywhere near that many. Some of her drawers are other stuff.
I would suggest using sterilite drawers (from walmart or hobby lobby with coupon) and stack. Label them by color and put everything in drawers by color. Designer paper could go in with the predominant color, and you could have a scrap of paper that lists the other colors in the paper too. These drawer units are smaller than the iris boxes and much more manageable space-wise.
I have seen a similar idea of Stacy Julian's where she keeps all embellishments...sequins, brads, ribbon, paint samples, rhinestones and pearls, and so on...in color drawers. Oranges, greens, turquoises, blues, reds, etc. Her papers are kept in paper sorters.