My mother is Guantanamera. She is, in fact from Caimanera, the cuban town next to Naval Station on Guantanamo Bay. My grandfather was a commuter worker, travelling daily from Caimanera to the Naval station for ~15 years
Guantánamo in Cuba is a city (Guantánamo City), the capital of a province: Guantánamo Province, where a Bay is located: Guantánamo Bay. So Guantánamo is not only the Naval Station.
The Naval Base is a portion of land on, or around the Guantánamo Bay, in Spanish we say “the mouth of the bay” where the bay connects to the sea, both sides of “the mouth” are currently part of the Naval Station, the rest of the Bay is Cuban territory as well as the rest of the province and the capital of the province, Guantánamo City are part of Cuba. And yes, cuban boats can sail from inside the bay to the open sea.
Guantánamo province has 6164 squared km, and the Naval Station has 49.4 squared km (the land part), so Guantanameros, even when not happy with the naval station, still have some space to live in.
Oh, the last time my mother (and I)visited her home town was around 1978–1979, I was pretty young and remember the fenced border a bit far from the road to Caimanera, and some other small details (the cementry), we were allowed to enter Caimanera Town because she was born there otherwise we wouldn’t be able to, at least at that time only persons born, or living there, or doing business there were able to enter Caimanera Town.