Georgia, Post-War Midcentury
You can basically Googlemap this story.
We know that the family begins in Atlanta (or the Atlanta suburbs) and that they travel a few hours south to the town of Toombsboro, where the grandma convinces her son to take a detour onto a dirt road. There's plenty of local flavor along the way: we see old plantations, Red Sammy's barbeque joint, and some sights out the window.
The second half of the story takes place in the ditch in the middle of nowhere where the family lands after running off the road. We're told the ditch is about ten feet below the road, and lies between the road and a "tall and dark and deep" forest. There's forest on the other side of the road too, so the forest "looms" menacingly over the scene on both sides. This part of the story is like a staged play: the site of the action doesn't move, the ditch is the stage, and the forest is "backstage," where characters are taken. We only learn what's going happening from the noises we hear (usually screams or gunshots).
As for the time, the era of the story is never explicitly defined, but given the cars and the mention of Gone With the Wind (published as a book in 1936 and released as a movie in 1939), we can guess it's the 1940's or later. Since there's no mention of a war going on, and the grandmother says that "the way Europe acted you would think we were made of money" (44), it's almost certainly after WWII. Since O'Connor wrote this story in 1953, we're going to place it in the late 40's or early 50's.
The particular timing of the story is a more interesting issue. We know that the family leaves their home in the morning, and that they leave Red Sammy's in the "hot afternoon" (presumably it's summer). We don't actually know how late it is, though, when they land in the ditch. The grandmother says it's a beautiful day, but we know from The Misfit that the sun is nowhere to be seen.