A Federation starbase to which Kirk intends to take the orphaned children of the Starnes expedition in the episode And the Children Shall Lead.
Starbase 4 is a Starfleet facility referred to in the original series episode And the Children Shall Lead, set in 2268. It is not shown.
After the USS Enterprise finds the adults of the Starnes scientific expedition dead on Triacus, apparently by their own hands, the surviving children are brought aboard. McCoy fears they are suppressing their grief, and Kirk decides to deliver them to Starbase 4, where they can be properly cared for.
The ship never arrives. Under the sway of the entity called the Gorgan, the children use the powers he has given them to cloud the minds of the crew, and the Enterprise is secretly diverted towards Marcos XII, a populous colony the Gorgan means to conquer. Only when Kirk breaks his hold over the children does the ship resume its proper course.