2268 (Columbia)

February

Stardate 5027.3: The crew of the Enterprise attempts to steal a Romulan cloaking device. (TOS 59: "The Enterprise Incident")
Stardate 5029.5: The crew of the Enterprise rescues a group of children stranded on a planet, along with their evil "imaginary" friend. (TOS 60: "And the Children Shall Lead")

Stardate 5121.5: While visiting a doomed planet, an Enterprise landing party is subject to torturous experiments to test an empathic race. (TOS 63: "The Empath")

Stardate 5423.4: An overpopulated race of aliens abduct Kirk to solve their population problem. (TOS 72: "The Mark of Gideon")
Stardate 5431.4: Captain Kirk pursues aliens who have stolen Spock's brain. (TOS 61: "Spock's Brain")

Stardate 5476.3: As Dr. McCoy discovers he is dying of an incurable disease, the crew of the Enterprise rush to stop an asteroid from colliding with a Federation world, only to discover that the asteroid is, in fact, a disguised alien vessel. They find an entire civilization living in the ship who believe they are actually on a planet, and a dictatorial "Oracle" who forbids any attempt to discover the truth. (TOS 65: "For the World is Hollow and I Have Touched the Sky")

Stardate 5630.7: The Enterprise transports an alien ambassador who must travel inside a special case, because his appearance causes insanity. (TOS 62: "Is There In Truth No Beauty?")
Stardate 5693.2: Captain Kirk is caught between dimensions while the Enterprise is trapped by an energy draining web spun by the Tholians. (TOS 64: "The Tholian Web")
Stardate 5710.5: Invisible "time-accelerated" aliens take over the Enterprise and attempt to abduct the crew for use as "genetic stock." (TOS 68: "Wink of An Eye")
Stardate 5718.3: Captain Kirk visits a mental health facility and confronts an insane starship captain who believes he is destined to control the universe. (TOS 71: "Whom Gods Destroy")

Stardate 5725.3: Strange, energy-based alien life forms threaten the Memory Alpha station and the Enterprise crew. (TOS 73: "The Lights of Zetar")
Stardate 5730.2: The Enterprise picks up the last two survivors of a war-torn planet who are still committed to destroying each other aboard the ship. (TOS 70: "Let That Be Your Last Battlefield")
Stardate 5784.2: The crew of the Enterprise encounters an ageless and mischievous race of psychic humanoids who claim to have organized their society around Ancient Greek ideals. (TOS 67: "Plato's Stepchildren")

Stardate Unknown: The crew of the Enterprise visits an abandoned outpost guarded by a mysterious computer. (TOS 69: "That Which Survives")
Stardate 5818.4: Captain Kirk races against time to acquire plague-fighting minerals from a world in the midst of a civil uprising. (TOS 74: "The Cloud Minders")
Stardate 5832.3: The Enterprise is hijacked by the criminal Doctor Sevrin and his loyal, hippie-like followers, who are attempting to find paradise. (TOS 75: "The Way to Eden")
Stardate 5843.7: The crew of the Enterprise encounters an immortal human who lives as a recluse on a planet of his own. (TOS 76: "Requiem for Methuselah")

Stardate 5906.4: Aliens force Kirk and Spock to battle illusionary villains in a test of good versus evil. (TOS 77: "The Savage Curtain")
Stardate 5928.5: Captain Kirk's consciousness becomes trapped in the body of a woman bent on killing him and taking over his command—while inhabiting his body. (TOS 79: "Turnabout Intruder")
Stardate 5943.7: Kirk, Spock, and McCoy are trapped in the past on a world threatened by a supernova. (TOS 78: "All Our Yesterdays")