Multiple realities
(covers information from several alternate timelines)

The 498th Air Base at night

Captain Christopher's base

Kirk and Sulu infiltrate the 498th Air Base

The 498th Air Base Group was a military unit located on an unspecified Air Force base near Omaha, Nebraska, United States of America in 1969 as part of the Strategic Air Command. In that year, the 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, Technician Webb reported the RADAR contact, and his superior informed Air Defense Command, which scrambled 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 Christopher had been assigned to the 498th Air Base in 1968, and was therefore able to provide the Enterprise crew with detailed information on its layout, including the location of the record section and the photo lab. (TOS: "Tomorrow is Yesterday")

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, Washington. The "Omaha Installation" referred to in this episode is therefore most likely Offutt AFB, which is located in the Bellevue suburb of Omaha, Nebraska. The base "housekeeping" unit in 1967 was the 3902nd Air Base Wing. Also, in 1967 Offutt was headquarters to the Strategic Air Command (SAC), which is precisely where such photos and tapes would have been taken. The colonel and security guards have the SAC emblem on their belt buckles and their berets, further implying that this is probably Offutt AFB, Nebraska.
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, although based upon time taken to intercept the Enterprise, it can be assumed that Captain Christopher was based at Richards-Gebaur AFB near Kansas City, Missouri.
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