I know that a runner in fair territory hit by a batted ball is out, but does this also apply if the runner is standing on a base at the time? Since the bases are all in fair territory and we are assuming a fair ball, I would suppose that the runner is out. A lot of people think that the base is a "safe" area.
Is this the right interpretation?

In baseball, a runner on or off a base, hit by a batted ball (in fair territory),is out, unless it has passed a fringe fielder (not counting the pitcher) 5.09f. The only exception is the runner is safe if he is hit by an infield fly while on a base (ball is dead)7.08f

You are right, this is one of those myths (#21, I think).