If i have an object with a defined mass, like a stone, and put it on the shelf it will sit there forever (lets forget all external influences).
But if I have to hold up the stone I will eventually get tired, feel pain, and give up.
Obviously I am using up energy to keep the stone where it is. But, if it takes energy to keep an object at a certain hight it could not sit on the shelf forever because it would need infinite energy.
What am I missing?
The energy being used is lost internally, in your arm.
Your arm is not a solid, it has joints around which it is free to move. The way to hold them in place is to have your muscles act against gravity.
Now muscles are not "solids" either: they are made with filaments which can slide relative to one another, these filaments are connected by molecules called myosin, which use up energy to move along the filaments but detach at time intervals to let them slide.
When you keep your arm in position, myosins hold the filaments in position, but when one of them detaches other myosins have to make up for the slight relaxation locally. They use up energy in doing so.
