Obnoxiousness is the noun: the quality of being very offensive, unpleasant, or rude. You use it when you want to name the thing itself — the loudness of the comments, the hardness of the smell — rather than call a person obnoxious. It is a little more formal than the adjective, and it still carries a sharp judgment. You can speak of the obnoxiousness of a remark, of a smell, or of someone’s manner. Same strong everyday insult, just turned into a thing you can point at.