The breakdown of forces would be something like this:
You are hanging down with your hands straight: Your hand is pulling down the bar with your weight, and your hand (and body which is attached to it) experiences upward force equal to that, keeping you from falling, acceleration is 0, so net force is 0
You pull yourself up: You accelerate up, so net force cannot be 0 anymore, you pull the bar (and consequently earth itself that is attached to it) down, while you go up
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