How accurately does the story portray the events?

It depends what you mean by "events."  If you are referring to events as they take place in Peyton Farquhar's head, then yes.  If you mean events as they take place in reality, then no.  In the moments before Farquhar is hanged for attempting to destroy the Owl Creek Bridge, he senses that time begins to slow down.  As a result, the moment between when he begins to fall from the bridge and when the noose snaps his neck is extended in his mind, so much so that he imagines an elaborate escape from the Union army, traveling through the forest for many miles, and, finally, a near-reunion with his loving wife.  However, time doesn't actually slow down; Farquhar only imagines it does.  He doesn't actually escape; he is hanged in real time.