What is "stuffy"/"stale" air composed of?
It may not be a complete answer but I can think of a few distinguishable features that may or may not be chemical reasons:


chemicals created by bacteria and organisms that are present in everyday households (mold, light activating some production from bacteria and micro-organisms, $NH_3, S^{2-}$ related compounds (they smell bad), etc). 
Compound this with the acknowledged lack of circulation and you have funky smelling air.
high humidity: how one can "feel" "stale, muggy" air.
high carbon dioxide concentration can make the air feel heavier (from interior design sources).