If you've got the time and patience, you might consider looking for a suitable used piece of furniture and refinishing it. I've done that with a number of pieces of furniture as winter projects the past few years and ended up with some very nice pieces of furniture that cost no more than a nominal amount to buy, and a bit of time to refinish them. I think you can end up with nicer than new pieces of furniture that way, and you'll appreciate it more when you've finished it.
You mentioned a chest of drawers -- 3-4 years ago I refinished a complete bedroom set of Cushman furniture (probably made pre-1950) for my youngest son and daughter-in-law, which included a chest of drawers. I think they paid around $200.00 for the entire set. The wood was Canadian white birch. Here's a picture of one of the pieces in the set that I refinished. There was another taller chest of drawers that would have made an even better storage places for feathers, and fly tying items:
You could easily add cedar linings to the inside of drawers to furnitures such as that, but I'm not sure it is necessary. It does smell nice though.
I've kept a lot of my furs and feathers in a cherry slant top desk for the past 40+ years. I just put the feathers, etc. in the drawers, and add a few moth balls from time to time and have never had any problems with moths or other insects. I also keep other materials in plastic containers of differing types, a large tackle box, and in other places with moth balls.