Runner on third base. Third baseman is playing "back" meaning the third baseman is further from home plate than is the runner. Batter hits fair ball which hits the runner while the runner is standing on third base. Is the runner out? Is the batter out?

Answer: The runner is out, the ball is dead, and the batter is awarded first base. Note: had the fielder been playing "in", meaning the third baseman is closer to home plate than is the runner, then the answer would be very different.

Under Official Baseball Rules Rule 7.08 Any runner is out when—
(f) He is touched by a fair ball in fair territory before the ball has touched or passed an infielder. The ball is dead and no runner may score, nor runners advance, except runners forced to advance. EXCEPTION: If a runner is touching his base when touched by an Infield Fly, he is not out, although the batter is out;
Rule 7.08(f) Comment: If two runners are touched by the same fair ball, only the first one is out because the ball is instantly dead.
If runner is touched by an Infield Fly when he is not touching his base, both runner and batter are out.