Excellent history and photo essay on Tristan de Cunha, the "remotest [inhabited] island in the world," population 265 as of 2016
Not a single ship visited Tristan da Cunha from 1909 until 1919, until the HMS Yarmouth finally stopped by to inform the islanders of the outcome of World War I. Accessible on by sea, Tristan da Cunha is in fact an archipelago, the remotest inhabited one in the world, although only the main island was settled by man, with a permanent population of 265 residents as of September 2016.
There is no airstrip; the island is accessible only by a difficult six-day boat voyage from South Africa.
They don’t have home internet, but they do have an internet cafe. Or did in 2016.