What are the rules in cricket if a batsman does not play a shot and the ball touches their pad? Can the batsman take a run?
No, unless the striker was trying to avoid being hit by the ball. This is made very clear in (the newly numbered) Law 23, but there is no substantial change from the old Laws:
23.2.1 If a ball delivered by the bowler first strikes the person of the striker, runs shall be scored only if the umpire is satisfied that the striker has
either attempted to play the ball with the bat
or tried to avoid being hit by the ball.
23.3 If in the circumstance of 23.2.1 the umpire considers that neither of the conditions therein has been met, then Leg byes shall not be awarded.
In particular, this means that you can't score runs by kicking away balls which pitch outside leg stump.