Just dumb luck?
 

Had a little league game where the batter hit a pop fly half way down first base line. Fielders weren't near it and it came down directly on the batter-runner.... i had to call her out because the ball touched her outside the box.... Just told her "that's just bad luck"...

Besides the box, is there ever an instance where the batter being hit by the ball in FAIR territory is allowed? I believe i recall a rule stating if the ball has gotten beyond a fielder (say F4) and strikes runner, it is live?

It's it the ball goes through a fielder or passed a fielder within reach AND no other fielder had a play on the ball.

To be an out the ball has to be fair AND have touched they runner in fair territory. In your play this really matters as the ball and the runner were near the line.