Surely the runner is in trouble if the doors close while she is inside. The pole is sure to get caught. You can't have it both ways. In SR, that events separated in space that appear simultaneous in one frame of reference need not appear simultaneous in another frame of reference. The closing doors are two such separate events.
SR explains that the two doors are never closed at the same time in the runner's frame of reference.
So there is always room for the pole. The far door is closed first. It opens before she gets there, and the near door closes behind her. Safe again - either way you look at it, provided you remember that simultaneity is not a constant of physics. If the doors are kept shut the rod will obviously smash into the barn door at one end.