It exists beyond time and space!
But seriously, in game? I believe it is as old as the Planes themselves, as old as all of Creation. I believe that the official explanation of the Far Realm is that it's the very edge of Reality, where the Planes start to bleed off into non-existence (Nihilo, or the Outer Dark, in my campaign), and Reality itself begins to unravel and peel apart in layers like an onion.
At least I really seem to recall reading that somewhere. Maybe I've gotten it confused with how I use it in my game, but I'm pretty sure that's the official description of the Far Realm.
And I do have Azathoth and all the other Outer Gods as coming from the Far Realm, beings born at the edge of reality, out of the crumbling edge of Creation.
Out of game, I think it's existence dates from the very last years of 2nd Edition.