Sound like, inspite of appearances, your barn does really not want to fall down. Maybe you should consider repairing it rather than demolishing it?

If you must, the pegs can easily be removed by drilling them out. However this must be done very carefully because your barn is no longer sitting plumb and square. As a result, many of the framing members may be in the opposite state of compression/tension from their original condition. Removing a single peg at the wrong point could result in a domino like collapse of a portion of the entire structure --- much like the failure of truss roof. It's not a good idea to be inside when this happens. A better idea is to try to remove a peg and a stick at a time rather than all of them at once.