The capacity to exert continuous force is just a property of rigid bodies.

Your body is using chemical energy to hold itself there because the human body isn't made to hold itself in the push up position--it has to use muscular energy to keep from collapsing because it's made of joints and tendons and shit that would collapse without muscles holding them in place. Except muscles have to use energy to hold things in place. You have to exert energy to hold yourself in the push up position, just like if you were lifting something off the ground. Which is the same thing.

Ugh.