Outpost 7 was an outpost constructed on an asteroid by Starfleet in 2160. This outpost was one of eight Earth Outpost Stations located in Sector Z-6 along the Federation side of the Romulan Neutral Zone. This zone was a strip of space that acted as a buffer between the Federation and the Romulans in the Alpha Quadrant. (TOS: "Balance of Terror", display graphic)
This outpost was never mentioned in dialog; this name came from a map of Sector Z-6.
According to the Star Trek: Star Charts (pg. 66), there were 23 Earth outposts. Outposts 1 to 8 were mentioned in dialog or were seen on the Z-6 Sector map in "Balance of Terror" and Outpost 23 was mentioned in dialog in "Future Imperfect". Outpost 7 was constructed in an area of space that been contested over during the Romulan War Theater of Operations (2156-2160).