Becoming a Barbarian when you started out something else is easy; it's called snapping. Something happens...you snap. Somebody dies...you snap. The world becomes a giant pesthole where every second somebody wants you dead...you snap.You're in the wilderness for 4 months with a guy who doesn't know how to tune his violin.
I never saw the barbarian as something a multiclass character aspires to do or to become, but as something they HAVE to become in order to survive, discarding (for the moment) their original way of thinking for sheer self-preservation.
Which works out better than taking that same scenario and multi-classing as a fighter where you would be trained in martial combat. Instead you would simply find the biggest thing you could find and hit someone with it as hard as you could until they die.
Like if you were a wizard stuck out in a place where you can't get spell components and you hadn't taken Eschew Materials yet. It's all about survival.