Starbase 27 was a Federation outpost that was administered by Starfleet in the mid-23rd century. Omicron Ceti III was located near this base.
On stardate 3417, the residents of the Omicron colony were evacuated to this facility by the USS Enterprise on the orders of Admiral Komack. (TOS: "This Side of Paradise")
Presumably, Starbase 27 was located in the Alpha Quadrant.
The unnamed planet
seen in TNG
" was identified as Starbase 27 in the script of the episode, but this detail was lost in the filmed episode.