The most common and most likely answer to this crossword clue is the 5 letter word FASTS
I've seen this crossword clue in The Daily Telegraph
F
A
S
T
S
Buddy explains!

The straight part
We know that "doesn’t eat" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case doesn’t eat = fasts.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 FASTS mean?
FASTS noun- A train that calls at only some stations it passes between its origin and destination, typically just the principal stations
- The act or practice of abstaining from food or of eating very little food.
- The period of time during which one abstains from or eats very little food.
- To restrict one’s personal consumption, generally of food, but sometimes other things, in various manners (totally, temporally, by avoiding particular items), often for religious or medical reasons.