The most common and most likely answer to this crossword clue is the 10 letter word PICNICKING
I've seen this crossword clue in The Sunday Telegraph
P
I
C
N
I
C
K
I
N
G
More clues leading to the result PICNICKING
Buddy explains!

The straight part
We know that "Eating out" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case eating out = picnicking.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 PICNICKING mean?
PICNICKING verb- To take part in a picnic.
- An expedition for the purpose of having a picnic.