Objectionable things are unwelcome or disliked. Your objectionable roommate might leave his dirty dishes everywhere, watch TV with the volume turned all the way up, and keep bringing home stray kittens.
If you've got a problem with someone's behavior, you can describe it as objectionable. Things like hate speech, bad jokes, and terrible music are all objectionable — obnoxious or even deplorable. In other words, you object to these things. The Latin root of object and objectionable is obiectare, "to oppose" or "to set against," literally meaning "to throw or put against."