[n3k5]

> And two people outside shut the doors simultaneously

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.