Also, please advise what you mean when you say "that wavelength can become very very large."...this sounds of great interest......and fear for us....as we dont want the ~3 metre waves to come out....they have given us radiated problems already
Correct answer is I do not know but like lesser mortals, I shall try.
1. The difference between a plastic and a metal, as far as EM shielding goes, is conductivity. And then, metals with higher conductivity can shield better. A superconductor is theoretically perfect shield at all frequencies.
2. At low frequency, an isolated metal box acts like an antenna and EM radiation can pass through thin metal bodies (via the famous skin effect). Even for superconductors, there is London penetration effect, but low and high (frequencies) depends on the object properties (wall thickness, conductivity and geometry). High frequencies are cut-off very effectively (like submarine radios while under water; the saline water is a conductor but nothing comparable to a metal). I do not have the equations with me ...
3. When you ground the box (bury it deep inside earth; this is the preferred but most inconvenient way) using a conducting wire, the impedance of the connection is a key factor. At high frequency, the impedance may be high enough and the box may behave as if it is not grounded. In your particular case, I do not know whether grounding (what is available at the power socket) will help at all.
4. In this particular case, your box does not act like a cavity (it is too small) because the lambda is around 3 meter. A large part of the EM radiation will be dissipated within the box itself and a possible solution is to help this radiation to dissipate efficiently (like the paint on the stealth aircraft). If the radiation is strong enough, it can also interfere with other parts of the circuit (better to kill it at the birth place).