Small room near rear of apartment (6)
I believe the answer is:
closet
'small room' is the definition.
(I've seen this before)
'near rear of apartment' is the wordplay.
'near' becomes 'close' ('close' can be a synonym of 'near').
'rear of' indicates one should take the final letters.
The final letter of 'apartment' is 't'.
'close'+'t'='CLOSET'
(Other definitions for closet that I've seen before include "Cupboard (to come out of)" , "Small room or wardrobe" , "Wardrobe in the US" , "Tall cupboard, wardrobe" , "American cupboard or wardrobe" .)