Is a batted ball fair or foul if it strikes a runner while he is on base?

It depends. If a runner is in foul territory, and is hit with a batted ball, before the batted ball reaches 1B or 3B, then it's foul. But if a batted ball hits a runner in fair territory, before passing an infielder other than the pitcher, time is called, the ball is dead, and the runner is declared out for interference. The batter is awarded 1B, but all other runners must return to the base they previously occupied.