The most common and most likely answer to this crossword clue is the 3 plus 1 plus 11 letter word RUN A TEMPERATURE
I've seen this crossword clue in Financial Times
R
U
N
A
T
E
M
P
E
R
A
T
U
R
E
More clues leading to the result RUN A TEMPERATURE
Buddy explains!

The straight part
We know that "Feel unwell" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case feel unwell = run a temperature.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.