But from the runner's perspective, the barn has length 5m long. What does she see happen when the doors are closed on her?
What one frame of reference sees as a simultaneous closure of the doors looks like non-simultaneous closure of the doors in another frame of reference.
Simultaneity means that two events coincide in time and space.
Different time and/or space means: not simultaneous!
No, they don't. I'd phrase it differently, e.g. “each door is shut for an instant when the end of the pole is just inside the door”. (Also, make it clear that the barn is long enough to fit a 10m pole between the two shut doors.)
When you put it as “shut the doors simultaneously”, it's more troll physics than a paradox.
Meta-puzzle: figure out how to still imply that the runner is completely enclosed for an instant, without incorrectly insinuating that it is so from her perspective.
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.
But I think over longer distances there would be a time delay for the light of the explosion to reach the individual, so they would probably the light of a visually warped pole colliding with visually warped barn doors very briefly before disappearing into an explosion.
Also, relevant xkcd (what-if) https://what-if.xkcd.com/1/
Most likely from the runner's perspective the world turns white as they dissolve to plasma prior to reaching the barn.