Old people’s home had no room, finally (4)
I believe the answer is:
'old people's' is the definition.
(ancient civilization of Edom)
'home had no room finally' is the wordplay.
'home' becomes 'ed' (I can't justify this - if you can you should believe this answer much more).
'had' is a charade indicator (letters next to each other).
'finally' suggests the final letters.
The last letters of 'no room' are 'om'.
(Other definitions for edom that I've seen before include "Kingdom of the Middle East" , "Biblical kingdom" , "In the Bible, the land of the people of Esau" , "Backward manner of existence" , "Biblical people" .)