Depends. For safety... yes. For the faraday to work as intended... no. Any charge will remain in the conductive material.So lets say someone built a faraday can. In this example, I'll say a 50 cal ammo can that is insulated inside with cardboard with a lid that has been sealed with foil tape. This would then be placed inside a cardboard insulated galvanized garbage can and the lid also sealed.
Would they need to ground the exterior of the garbage can with a copper wire tied to a ground rod as well, or leave it un grounded?
In your proposed setup you're looking at something like a -120... -140dB attenuation. As long as you're going to that much effort then why "not" run a ground?