are there any constraints on material used and wall thickness?
The only thing is that the mesh be very fine, and that there are no 'leaks' where EMI/RFI can get through. You can ground the box to existing plumbing pipes or electrical conduits. You will need to have a door to get in/out of the box, so there should be some contact 'fingers' to make a good electrical connection. Construction can be as simple or as eloborate as you want it to be, which can be a simple wood frame with the mesh stapled to it to hold it on, to a complete construction project with concrete walls and steel doors. Remember to do all sides (floor, ceiling, and all four walls) in conductive mesh, and bond them electrically together.
As a quick aside; if you don't ground the mesh correctly, it will become a really big inductively-coupled antenna.
Any power coming into the box (lights, power for the electronic devices, test equipment, etc.) needs to be filtered HEAVILY! Lighting should not be gas-filled (neon/flourescent) tubes or bulbs, as they create EMI/RFI.
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I think what you want is a Faraday cage. a cage or enclosure built of wire mesh or other conductive material. I'm not an expert on there construction but they could be used to create a RF free area or protect from EMP. I think you need the cage to either be on the ground or a complete cube and it must be tied to earth ground (electrically).