If something is unexceptionable, don't bother trying to find something wrong with it — you won't. Your unexceptionable character makes you the perfect candidate to run for public office, but if you run, people will be looking for things to criticize.
If you break it down, exceptionable describes something objectionable or unacceptable. Put an un on it, for “not,” and unexceptionable is something no one can object to. If your baking is unexceptionable, your muffins are light and fluffy and melt in your mouth. In this arena, even a top baker would be unable to find fault. As for your run for public office, if your character is truly unexceptionable, you won’t mind those reporters snooping around!