The 498th Air Base at night
Captain Christopher's base
The 498th Air Base Group was a military unit located on an unspecified Air Force base near Omaha, Nebraska, United States circa 1969. In that year, USS Enterprise arrived in low Earth orbit above the base following a nearly catastrophic encounter with a black star which had displaced it in time.
Meanwhile, at another unnamed Air Force installation, Technician Webb reported the radar contact, and his superior ordered Captain John Christopher, a fighter pilot, to intercept what was then believed to be a UFO. In one version of history, Christopher's aircraft was caught and crushed by the Enterprise's tractor beam. Captain Kirk and Lieutenant Sulu infiltrated the 498th Air Base to retrieve the photographic and audio records of the incident that were recovered from the wreckage of Christopher's aircraft. Later, Spock devised a way to replay history that effectively erased the Enterprise's presence altogether. Its brief appearance was doubtless logged as one more strange aerial incident.
Captain John Christopher had been assigned to the 498th Air Base approximately one year prior to the events of this episode, and was therefore able to provide the Enterprise crew with detailed information on its layout. (TOS: "Tomorrow is Yesterday")
Background
In 1967, there was no such installation as the 498th Air Base, nor was there a unit designated 498th Air Base Group. There was however, a 498th Fighter Interceptor Squadron, but it was based out of Paine AFB near Everett, WA. The base referred to in this episode is therefore most likely Offutt AFB, which was/is located near Bellevue, NE. The base "housekeeping" unit at the time was the 3902nd Air Base Wing. Also, in 1967 Offutt was headquarters to Strategic Air Command, which is precisely where such photos and tapes would have been taken. The footage shown onscreen was that of Offutt AFB, and more specifically of Building 500, the actual headquarters building of the Strategic Air Command.
Given the Enterprise's altitude at the time radar contact was made, Captain Christopher's and Webb's base could have been anywhere in North America.