The most common and most likely answer to this crossword clue is the 10 letter word ECCENTRICS
I've seen this crossword clue in Financial Times
E
C
C
E
N
T
R
I
C
S
Buddy explains!

The straight part
We know that "Odd characters" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case odd characters = eccentrics.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 ECCENTRICS mean?
ECCENTRICS noun- One who does not behave like others.
- A kook; a person of bizarre habits or beliefs.
- A circle not having the same centre as another.
- A disk or wheel with its axis off centre, giving a reciprocating motion.