The most common and most likely answer to this crossword clue is the 7 letter word EYELIDS
I've seen this crossword clue in The Guardian
E
Y
E
L
I
D
S
More clues leading to the result EYELIDS
Buddy explains!

The straight part
We know that "They go down for a sleep" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case they go down for a sleep = eyelids.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 EYELIDS mean?
EYELIDS noun- A thin skin membrane that covers and moves over an eye.