Because we're talking about a story very much rooted in history, it's important to think about the historical setting. In the story itself, Bierce never tells us the date of the famed occurrence at Owl Creek Bridge. Though the first version of the story included the phrase "summer of 1862," subsequent versions did not (source) – and if it didn't make the final cut, we can't use that as evidence. What the narrator does tell us is that the Confederate army has recently suffered setbacks and that Corinth has fallen. Aha! Evidence of the American Civil War. Looking at our dusty history books (or this handy website), we learn that the Confederates pulled out of Corinth, Mississippi in late May of 1862 when the Union invaded. So, even without the original date marker, we can guess that the story takes place in the summer of 1862 (or maybe late spring, if you want to be a stickler).
Now that we have the date set, what does that mean for our story? Well, in the summer of 1862, the Civil War has been in full force for about a year and it is far from over. Alabama, home to Owl Creek Bridge, has already seceded (pulled out of the Union) and its citizens are largely committed to the Confederate cause. That's all to say that we're in a majorly Confederate area. When the story begins, Union troops have entered Alabama. Peyton Farquhar, a civilian slave owner, is determined to serve his newly formed country.
Bierce goes into great detail about the bridge and the surrounding landscape, and he uses his protagonist's heightened senses to observe and describe this setting. Though Farquhar is a native of this part of northern Alabama, his journey home takes him through a seemingly unfamiliar landscape. A forest in Alabama becomes a threatening, eerie dreamscape as Farquhar moves toward death. So, throughout the story, we have a strange mix of detailed reality and imaginative fantasy. Before we reach the revealing ending, it's hard to determine which is which.