They were from a rural Southern state that smug Hollywood scriptwriters liked to mock. The actual name of the state is ultimately unknowable.
As an aside, the world would not be safe if you applied your talents to evil instead of good
Didn’t we just go through this a couple of months ago?
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But the area where the three states converge is very flat delta land. There may have once been a town called Sibley in this area but I have not considered it because it could not be considered “in the hills” by any stretch of the imagination.
It’s Ar-kansas.
It’s a hell of a town.
Nice research. I always thought it was Tennessee while watching the shows in syndication
While reading the OP, Did anyone get flashbacks to Lion vs Tiger?
I saw a similar discussion about what state Benson was in. I would think that the dialog or story lines in a fictional place is based on what fits what is in the episode. If they want mountains 100 miles away then that’s where they put them.
Nah, Danville. Danville, AK, to be exact. 
Ynnad, not to make light of your OP. I always pictured Kentucky, FWIW. (I know, nothing).
For some reason I was certain that it was Minnesota.
Soap, the series where the character first appeared, was set in Connecticut.
Well, the first thing you know, old Beck’s a millionaire…
I cannot find Danville, Alaska anywhere in your Wikipedia link. Did you mean Danville, Arkansas (Danville AR)?
To find Jed’s old cabin you start out near Dardanelle in Yell county. Take the last train to Clarksville and then head north. It’s a fur piece up there but somewhere around Sand Gap you’ll find a trace headed west, take that. Not really a road, more of a hog waller, but if you take your old blue tick hound and look around, you can find it.
I’m sure you have to turn left at Greenland.
I always thought Arkansas but I remember in one episode how Granny missed snow. Seems to me there’s more snow in the hills of Tennessee than Arkansas.
Specific nearby towns- you’ll never nail it down. As the OP’s copious notes show, continuity was not a thing in 1960s sitcoms.
Unfortunately most of those old rural sitcoms were cancelled in the great Rural Purge of the early '70s. Yeah, it was a real thing. Even though many of the shows were still popular, they were cancelled in favor of shows for a younger market.
That is how we got a decade or so of city cop shows and other drama.
I don’t remember who said it, but they got rid of everything that had a tree in it.
It was the actor who played Mr. Haney on Green Acres.
Good article here:
Chi-town?