But gaining access to swing sets temporarily used by other shows might have been more difficult, and those other shows would have had priority in airing those sets first. Using standing sets from other shows, like Jim Phelps' apartment in Mission: Impossible, probably wouldn't have been allowed.
Actually it was allowed and it was done. There's a well-known example of a
Mission: Impossible episode that was shot on a redress of the
Brady Bunch home set:
http://www.invisiblethemepark.com/2013/12/when-mission-impossible-invaded-the-brady-bunch-house/
And then there was the second-season M:I episode "Trial by Fury," which was shot on the part of the Culver City backlot used as Stalag 13 in
Hogan's Heroes. And of course there were Kirk and Edith walking down a Mayberry street and passing Floyd's Barber Shop in "City on the Edge of Forever." And I think I remember reading once about a
Get Smart episode using
Gilligan's Island's lagoon backlot. If shows could use each other's regular backlot sets, there's no reason they couldn't use each other's soundstage sets as well. It's just a matter of scheduling.
There are also instances where one show is able to use sets that were made for another production that's recently ended. There was a
Sliders episode that made use of the standing sets from the short-lived
Timecop series. And the interiors of the battlestar
Pegasus from the
Galactica reboot were redresses of the sets for the failed
The Robinsons: Lost in Space pilot. And when
Stargate SG-1 did episodes featuring William Devane as the US president, they were reusing the White House sets built for
X2: X-Men United. It happens all the time.
Unlike GM's suggestion, these do all have a consistent central character.
Yes, obviously, but the point is that they were still trying to approximate anthology-style storytelling despite having a consistent lead. The advantage of having a regular star was that it brought that star's fans back from week to week, and it was probably cheaper to have a regular under contract then to hire a new cast each week. But at the time, the preferred approach to storytelling was the anthology model. In the '50s, the classiest TV shows had been the anthologies, the "playhouse" series that adapted a different pre-existing or original play each week, and so that was the model that other shows aspired to creatively, even when they embraced the logistical and production benefits of a continuing lead. That's why we've had so many shows where the lead was dealing with different guest characters' problems every week, or where the lead was adopting a different identity every week. Following a continuing lead like Richard Kimble or David Banner or Sam Beckett was just an excuse to meet a different set of characters with a different set of problems in each new episode. It was one step above an anthology series having a regular host like Rod Serling or Alfred Hitchcock. There was a radio anthology in the '40s called
The Whistler where the title character was nominally involved in the situations he encountered each week, but in practice was little more than a spectator and narrator for stories that were about the other characters, with the Whistler himself remaining a cipher. Since characters like Kimble and Beckett were trying to subsume themselves within their adopted identities, the stories weren't really about them most of the time. They were about the people they encountered, a different drama with a different cast every week. The dramatic benefits of an anthology combined with the production benefits of a continuing lead.