Sed works on a line basic, and a,b{action} will run action for lines matching a until lines matching b. In your case
sed -n '/aaa/,/ccc/p'
will start printing lines when /aaa/ is matched, and stop when /ccc/ is matched which is not what you want.
To manipulate a line there is multiply options, one is s/search/replace/ which can be utilized to remove the leading aaa and trailing ccc:
% sed 's/^aaa\|ccc$//g' /tmp/test
bbb
Breakdown:
s/
^aaa # Match literal aaa in beginning of string
\| # ... or ...
ccc$ # Match literal ccc at the end of the sting
// # Replace with nothing
g # Global (Do until there is no more matches, normally when a match is
# found and replacement is made this command stops replacing)
If you are not sure how many a's and c's you have you can use:
% sed 's/^aa*\|cc*$//g' /tmp/test
bbb
Which will match literal a followed by zero or more a's at the beginning of the line. Same for the c's but just at the end.