Use a 1/8" square or fishtail endmill to mill the walls for the box set. A 3/4" or taller material will be best for making a functional box. Use double sided tape to keep the nesting sections from shifting
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Choose a 1/8" material to use as your lid. Expanded PVC cuts quickly and comes in good colors. A translucent acrylic also works really well.
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Mill the box floors from either the same material as the top, or choose one that compliments it well.
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If everything cuts correctly, no glue is required for assembly. The base fits into the side with the small ledge cut into it. For extra durability you can squeeze a small amount of super glue into this area.