>Are they self-sustaining? How do they pay for stuff the want to import
Generally the modern day population of these types of islands are simply cover for the government to maintain political control of an area of ocean surrounding them. Same deal with the Falklands, Orkney/Shetland, etc. To that end their entire existence is more or less subsidised because of this.
Orkney is only 20 miles from the UK mainland so I'm not sure that's the reason. People there make fudge which is pretty good. Until recently (1956) it hosted a major Royal Navy base.
Isn’t the hosting of a naval base a good example of this island being subsidized because the government wants to control the waters around it? And it being close to mainland is another reason, you don’t want another country having an island that close to you.