Does bending your arm in space require any energy?

The short answer is yes, bending your arm in the weightlessness of space still requires energy. You are correct that bending your arm does not require us to overcome the weight of your arm, but we do have to overcome its inertia. Inertia refers to the sluggishness that massive objects have (even in the weightlessness of outer space) given how much matter they have.

A bit of explanation to clarify things: we could just as well ask does it take any energy to push a 100,000 lbs. asteroid a distance of five feet while in the weightlessness of space?

I'm sure you'd agree that the asteroid doesn't weigh anything at all in outer space. But that doesn't mean you can just float up and thump it with your finger and expect it to take off at near light-speed. You still have to push on it (and expend energy in doing so) to overcome its inertia.

The simple summary of all of this is that when we try to move objects in outer space, we don't have to fight against any gravitational weight, but we still have to overcome the basic sluggishness that massive objects have (i.e. inertia).

Well, yes.

Movement of your body parts (hands, legs eyelids, etc.) occurs due to the contraction of muscle fibers. This process requires energy (from cleavage of ATP molecule to form ADP). This is the only way an astronaut can move his arm.

Transforming the internal energy (chemical) into mechanical energy requires the expenditure of ATP. So, the answer is yes.

But, you can make it smaller, I guess.

The arm has mass, even if it is weightless due to being in microgravity.

And force = mass • acceleration and work = force • distance.

For a mass to move, work has been done. Weight doesn’t play into this calculation at all.