From the frame of reference of the runner the pole will first hit the door at the end of the barn (unless it opens in time), and the door at the front of the ba

From the frame of reference of the runner the pole will first hit the door at the end of the barn (unless it opens in time), and the door at the front of the barn will only close just after the back of the pole has passed the door. If the doors actually don't open in time then the resolution for the fact that the pole doesn't physically fit is that nothing can be rigid at relativistic speeds. So either the door or the pole or both will deform (violently) to allow the pole to fit inside the barn.