The computer center was a section of Starfleet facilities responsible for the coordination of all information storage and transmission activity.
In 2267, Commander Spock sneaked into the computer center of Starbase 11 in order to forge a voice transmission to the USS Enterprise as part of a plot to commandeer the ship. Spock was caught by Chief Humbolt, a technician assigned to the computer center, who attempted to stop him, but Spock incapacitated him with a Vulcan nerve pinch. (TOS: "The Menagerie, Part I")
Sometime in the 2270s, Leonard McCoy didn't care for the sickbay of the refit Enterprise, saying it was "like working in a damned computer center." (Star Trek: The Motion Picture)
In the final draft script of "The Menagerie", Starbase 11's computer center was described as "a complex communications and library-computer center, the automated heart of the Star Base."
The computer center as seen in "The Menagerie" was a re-dress of the Enterprise engineering set.