while IFS= read -r line; do echo ${#line}; done < abc.txt
It is POSIX, so it should work everywhere.
Edit: Added -r as suggested by William.
Edit: Beware of Unicode handling. Bash and zsh, with correctly set locale, will show number of codepoints, but dash will show bytes—so you have to check what your shell does. And then there many other possible definitions of length in Unicode anyway, so it depends on what you actually want.
Edit: Prefix with IFS=
to avoid losing leading and trailing spaces.