I like cigar boxes.
Box to Store Chess Pieces
I like cigar boxes.
Yup. Cigar boxes are cheap if not free from your local cigar store.

Doubtful you will get one for under $30.
I like boxes from Hobby Lobby, something like these:
https://www.hobbylobby.com/search/?text=leather+box
The single boxes that hold a chess set are under $30.

You can buy a cheap box that is used to store cigars. They're called humidors I think. Many of them have a see through glass on top also. There really is no reason to purchase a box that is made "specifically for chess pieces"

earlier forum post on the same topic.....may provide some ideas...for example, eBay...
https://www.chess.com/forum/view/chess-equipment/chess-piece-storage-box-that-is-well-made-amp-reasonable
found this wood box by searching for "storage box" here....looks about the right size for chess pieces....comes in two colors - natural & dark brown....
https://www.amazon.com/Blake-Lake-Wood-Stash-Rolling/dp/B077VY2WJK/ref=sr_1_56?crid=2D52IJPSRTHB&keywords=storage+box&qid=1551264914&s=gateway&sprefix=storage+box%2Cstripbooks%2C209&sr=8-56
Sliding Sheesham Chess Box.....
https://www.wholesalechess.com/shop/chess-sets/chess-boxes/sliding-sheesham-chess-box
I've used these cardboard chess piece storage boxes.....inexpensive, functional, with individual compartments for the pieces.....
https://thechessstore.com/the-chess-store-box/
I searched Treasure Boxes and Leather Boxes and found many on Amazon that looked good and of a decent size.

You can buy a cheap plastic gun case for under $30, which is filled with foam, so when you lay the pieces down they get squished by the foam and don't move around. Some of them are 2 layer (for 2 or 4 pistols), so you can put white on 1 layer and black on the other. I use one to store my Isle of Lewis pieces which are cast and would chip if they banged around in a regular box. They come in all different sizes so you'd have to lay your pieces out and get an idea of how much space you need. If you have a Murdochs or Tractor Supply near you they have them there.

I just bought a pretty nice set of wood pieces. Can anyone recommend a box or other storage option under $30?