That's a helluva closet...You made me google for this?
An anechoic chamber (an-echoic meaning non-echoing or echo-free) is a room designed to completely absorb reflections of either sound or electromagnetic waves. They are also insulated from exterior sources of noise. The combination of both aspects means they simulate a quiet open-space of infinite dimension, which is useful when exterior influences would otherwise give false results.
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If you want it to be anechoic down to the lowest audible frequencies then the treatments will take up more volume than the closet.
The good news is that you probably don't really need an anechoic chamber. The bad news is that the closet is probably still too small for good recordings.