The Federation starbase to which the Enterprise is ordered to carry the colonists of Omicron Ceti III in the Original Series episode This Side of Paradise.
Starbase 27 is mentioned in the Original Series episode This Side of Paradise. The Enterprise is sent to Omicron Ceti III expecting to find the Sandoval colony dead from berthold rays. The settlers are alive and in perfect health instead, and Starfleet orders Kirk to evacuate them to Starbase 27, where they can be examined and resettled.
The colonists refuse to go. They are protected, and kept contented, by alien spores that soon infect the Enterprise crew as well, until the whole ship's company has beamed down and Kirk is left alone on board.
Kirk finds that strong emotion destroys the spores. With Spock he frees the crew and the colonists, and Elias Sandoval at last agrees to leave and asks to start the colony again on another world.