Sector 78 was a region of Federation space, a sector located near the Klingon Empire in the galaxy's Alpha or Beta Quadrants, and was a location on star charts used by the Federation in the 23rd century.
In 2270, the starship USS Enterprise was assigned to investigate an armed, automated vessel that had been reported in Sector 78, as Starfleet suspected that it might be a Klingon incursion. On stardate 6013.4, the Enterprise was in the sector, near or in orbit of the planet N'cai.
Three months later, a temporal disruption threw the Enterprise back to stardate 6013.4, but to a different location. Starbase 7 checked to see if there were now two Enterprises, but scanners showed that the Enterprise in Sector 78 had disappeared. (TOS - Mudd's Angels short story: "The Business, As Usual, During Altercations")