"I punch Mike" (Mike being a party member if no one else is available). The Barbarian is now in combat, rolls initiative, and can rescue his party from the flood.
Well, he would still have to make an attack or take damage every round. Mike might not be too happy with that.
Maybe Mike is also a barbarian, and now he's raging too.
My ruling would simply be: if you are not in an initiative situation, you cannot rage. Further, while this is probably a more grey area for many people, I tend to take the "rage" part more literally, in that you cannot think rationally and plan complex stuff while raging. I would not let you rage in the situation you described, but I may let you rage if, say, there is a door you cannot seem to budge, so you lose your cool and rage and "Hulk smash" it. Because that is what a raging barbarian has always been to me: Hulk smash.
Of course, but it has to come with both advantages and disadvantages.
E.g. Think about the barbarian in the tavern getting enraged, because his beer does not arrive in time. So the waiter might be intimidated into bringing it faster, otoh the table might have to survive a crushing blow (saving throw DC 10+ the barbarians strength) while the barbarian is hitting it.
Well, he would still have to make an attack or take damage every round. Mike might not be too happy with that.
Did the DM have the flood deal HP damage to your character?