Captain John Christopher had been assigned to the 498th Air Base in 1968, and was therefore able to provide the Enterprisecrew with detailed information on its layout. (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, WA. The "Omaha Installation" referred to in this episode is therefore most likely Offutt AFB, which is located near the Omaha suburb of Bellevue, Nebraska. The base "housekeeping" unit in 1967 was the 3902nd Air Base Wing. Also, in 1967 Offutt was headquarters to the Strategic Air Command, which is precisely where such photos and tapes would have been taken. The footage shown on screen 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.