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#55: Star Trek: The Original Series, season 2, episode 26 "Assignment: Earth"
Written by Gene Roddenberry & Art Wallace, directed by Marc Daniels.
Stardate unknown (August 2268)
"Very groovy." - Roberta Lincoln
The Enterprise uses the slingshot maneuver to travel back to 1968 to conduct observations of Earth during the 20th century. An incredible power surge hits the ship and a 20th century-looking man and a house cat appears on the transporter pad. He introduces himself as Gary Seven and, even though he came from an impossible distance away from the ship, he claims to be human and from the year 1968. Once Gary realizes that Kirk and his ship are from a different time period, he demands to be returned for fear of altering the timeline. Unsure of what to believe, and because Gary tries to overpower them, Kirk places Gary in the Brig while he tries to verify his story. Almost immediately Gary escapes with the help of a futuristic device and continues down to Earth. Kirk has no choice but to pursue Gary Seven down to Earth to stop him from doing whatever it is he intends to do.
Originally released: 3 May 1968
"Assignment: Earth" on Memory Alpha (mirror)
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What did you think?
In hindsight, the dark truth of this episode is that they knowingly travelled back in time on a proper mission, so it had to be Section 31 shenanigans.
They all got their memories wiped when they came back.
And, yeah, it would have really worked as a show then and probably would work as an alt-history piece now.