The most common and most likely answer to this crossword clue is the 6 plus 6 letter word NARROW ESCAPE
I've seen this crossword clue in The Guardian
N
A
R
R
O
W
E
S
C
A
P
E
Buddy explains!
The straight part
We know that "A Close Shave" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case a close shave = narrow escape.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.