I guess it is fair to ask what you define as human existence. Are you including all of the human family tree? If you are, then one of the early humanoids (I think it was the Cro Magnons) lived in harmony with the rest of their population.
If you mean just homo sapiens then the numbers are 3400 years of recorded history, 268 years of no war.
Furthermore you need a definition for war. The New York Times, in a landmark opinion piece, defined war as 1000 casualties.
I actually don't believe the 268 years statistic. The reason is whoever made the estimate, refers to recorded history. What about those places that did not have a written language? Surely some of them were at war during the 268 years. Anyway, there's room for doubt. And going back 3400 years does not seem like long enough. Events in Genesis go back much further than 1400 BC. You could question how many people were lost before the advent of Moses so 1000 people may not have been lost in any conflict you could point to like Joseph for example.
I can't confirm the assertion at the beginning of my answer, but excavations have found the two (Cro Magnon and Neaderthal Man in somewhat the same area -- that too is not from a source I would trust completely).
