The arms are actually very simple! It's just a few RigidBody2D objects connected by springs. The shoulder is fixed in place and the hand is attached to the mouse. They're connected with a Line2D object.
Something like this (but I didn't get the smoothing to work): https://www.reddit.com/r/godot/comments/1dlfve1/how_to_make_a_smooth_physicsbase...