It exists
The Quantum Barbarian hinges on the Berserker option (which causes him to start raging below a certain amount of hit points, and stop above that amount) and the Bear Warrior prestige class. Bear warriors turn into bears when they rage, gaining a certain amount of hit points in the process that they lose when the rage ends. As a result, if the berserker dips into their rage threshold within the amount given by Bear Warrior, they get caught in a cycle of rage-start-rage-end that won't end until or unless their HP drops below or above that threshold.
Beserker Strength (PHB2): When HP falls below Barbarian level x5, you enter rage. There is no limit to how many times this can happen. This rage can only end when you fall unconscious, helpless, or receive healing to bring your current HP above the threshold.
Bear Warrior (Complete Warrior Prestige Class): When you rage, you can choose to enter a bear form. This provides some extra bonuses, the important one being a +4/+8 (depending on level) CON bonus.
So:
- You fall below the HP threshold, and rage activates.
- As a Bear Warrior, rage increases your CON.
- This increase in CON raises your HP above the limit, so rage ends.
- Rage ending removes the extra CON, lowering your HP below the threshold.
- Go To 1.
As mentioned, this wouldn't work by RAW due to the part about rage ending when you get healing to raise you above the limit. A CON increase is not technically healing, so #3 in the above sequence wouldn't happen by RAW. It's also worth noting that Berserker Strength is technically not "Rage" and thus may not qualify for Bear Warrior at all (though your DM may not get fussy over that detail).
No, a barbarian does not need to be in combat to rage.
Defining things as "Actions", "Bonus actions", "Reactions" etc. are only to specify how much you can do in a turn or round of combat, since you effectively only have a 6-second timeframe to do things. Much like casting spells that require a minute long casting time - you can still attempt to cast them... you just need to be uninterrupted for 10 rounds.
The only catch is that to sustain your rage, you need to attack (which doesn't have to be successful, mind you), or take damage, "every round", so a timeframe needs to be defined outside of combat to sustain it:
Your rage lasts for 1 minute. It ends early if you are knocked unconscious or if your turn ends and you haven't attacked a hostile creature since your last turn or taken damage since then. You can also end your rage on your turn as a bonus action.
Note: the rule does state "hostile creature", so it'll be up to you and the GM to define what states as "hostile". Spanking a misbehaving child, or punching a passer-by may not exactly qualify.
However, as suggested in the comments, setting yourself on fire would be an ideal way to maintain the rage - constant damage, therefore, constant rage.
Best Answer
Maybe
The rage feature only states that "you fight with primal ferocity". The player could describe this in a number of ways: The character could launch him/herself into battle screaming maniacally or fight with a cold focus that hides an inner boiling anger.
I believe how obvious this is would be up to the DM, but bear in mind that concepts such as class and level are merely tools to enable the rules and have no meaning in the game (at least as directly relates to appearance).
If some person isn't wearing heavy armour (and a fighter may not be wearing heavy armour due to preference, affordability or because they are attending a royal banquet) then how do you tell if they are a fighter or a barbarian? If they are wearing heavy armour then how do you tell if they are a fighter or a cleric (unless they cast a spell)? And so on.
Of course, keeping clear of a very angry or focused individual intent on cutting you in half with an axe is possibly a wise course of action in any case.
Other features may be more obvious in-game: A barbarian's reckless attack is clearly representative of the character going 'all-out' but leaving themselves open to attack in return. But it would be a DM call (possibly based on the intelligence of the opponent) on exactly how that affects tactics without getting "meta"