A shelter; old, broken-down, at the back of the garage (5)
I believe the answer is:
lodge
'a shelter' is the definition.
(I know that domiciliate is a is a more specific form of the action of lodge)
'old broken-down at the back of the garage' is the wordplay.
'broken down' indicates an anagram.
'at' says to put letters next to each other.
'the back of' says to take the final letters.
'garage' becomes 'g' (this could be a standard abbreviation which I don't know about).
The final letter of 'the' is 'e'.
'old' anagrammed gives 'lod'.
'e' after 'g' is 'ge'.
'lod'+'ge'='LODGE'
Can you help me to learn more?
(Other definitions for lodge that I've seen before include "Provide quarters" , "Beaver's nest" , "Masonic association" , "Provide accommodation for" , "Stay temporarily, with bed and board" .)