Old people’s home had no room, finally (4)
I believe the answer is:
edom
'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'.
'ed'+'om'='EDOM'
(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" .)