Supermarkets and kitchen shops sell small screw top plastic boxes. Some clip lock boxes can also be found in smaller sizes. Old 35mm film containers will store about 4 or 5 of the smaller dies.

Personally I avoid wood. I’ve seen too many old tools go rusty due to moisture in the wood. A much better alternative is a plastic box lined with closed cell foam. You can easily make the cutouts in the foam for the dies with a pipe. Even a piece of suitably sharpened plastic pipe will cut foam. Cheap sources of foam are things like gym mat tiles, often sold in Aldi/lidl.

Might be worth asking someone like Tracy Tools.

Dave

3D printing a suitable case may be a solition:

This one is from Printables – Tap & Die Case

John

The Tap And Die Company used to advertise taps and die sets in British-made boxes, so presumably they bought in foreign items, possibly in the boxes you want.  They were taken over by C R Tools in 2023.

Let us know how you get on. It may be useful for someone else.