runner hit by batted ball
 

Is the ball always dead and the runner out even if no defensive player can make a play on the batted ball?

Short answer, no.

Long answer, since you used the word "always," the answer is no, but due to your qualifying clause regarding playing the ball, the answer you seek probably requires you to be more specific.

When a runner is hit by a fair batted ball, the runner is not out and the ball stays live if:

1. the ball has been deflected by a defensive player
2. the runner is directly behind a fielder and no other fielder has a play on the ball
3. a fielder could have played the ball but chose not to

There are probably others, but I can't think of any right now.

4. The runner is hit in foul territory (e.g., ball bounces off third and hits R3)
5. Some code other that OBR is being used. ;)
6. It's an infield fly, the runner is on the base, and does not intentionally interfere.

Situation: R1 on 1st, F3 holding runner on. For whatever reason F4 near 2nd base. Batter hits ground ball in huge gap between 1st and 2nd base which contacts R1. F3 or F4 had no chance to get to ball to make play. Is runner out? Thanks

In OBR, yes.

How about NFHS

If the ball has not passed F4 or F3 when it hits R1, R1 is out. If not, play on.

If the ball has passed an imaginary line between the two fielders and no other play was possible, the runner is not out.

The "string theory" is still alive?

Used in FED.

At least partially used in NCAA, I think.