The most common and most likely answer to this crossword clue is the 9 letter word ANTEROOMS
I've seen this crossword clue in The Guardian
A
N
T
E
R
O
O
M
S
Buddy explains!
The straight part
We know that "Chambers" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case chambers = anterooms.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does ANTEROOMS mean?
ANTEROOMS noun- A room before, or forming an entrance to, another; a waiting room.