Re-fluff the rage. Razor Fighting Focus, zen-state martial arts, what have you. Even if you want to keep it as hyper aggression, it could be by possession, by sharing your soul with an ancestor or spirit of an animal.
Just like any character, come up with an interesting story and work out from there. The mechanics of rage can fit a lot of different narratives.
And while the class is called "barbarian", there's no need to keep it as an uncivilized savage. Pit fighter, drunken sailor, whatever - even more with re-flavoring rage.
Exactly! The game mechanics of the class are not a role-playing straitjacket forcing you to be a loincloth-wearing savage Conan clone!
Be a super-civilised, highly trained warrior with anger management issues.
Take the Acolyte background and fluff the barbarian mechanics as a secret fighting style involving faith and focus.
In the Deathstalker books by Robert R Green, the Deathstalker clan is a noble house in a star-spanning futuristic empire. Each noble house had its own shtick, and the Deathstalkers had surgically implanted organs and other modifications that could release adrenaline and other engineered enzymes that temporarily increased their physical capabilities in combat, resembling D&D's Rage mechanic. They even had a limit of how many times they could 'rage' in a day, because it takes time and rest for the body to replenish those enzymes.
Or, y'know, just fluff some werewolf heritage into your family history.
All you have to do is divorce the game mechanics of this (or any other) class from any pre-conceived fluff. One way to get your mind thinking along these lines is to choose a background that the stereotypical Conan clone would never choose. If you choose Outlander, you're doing it wrong. Instead, choose Acolyte, Noble, Urchin, Folk Hero, Charlatan....whatever, and think about how the Rage mechanic could be fluffed for that ability to arise from that background. Once you've done that, your characterisation is half done, and you'll have a cool and refreshingly unique 'barbarian', without a loincloth is sight!