Can the Aland Islands, Faroe Islands, or Greenland compete in the Eurovision Song Contest? What about the Channel Islands or the Isle of Mann? Gibraltar?
It’s organised by the European Broadcasting Union, an alliance of public service broadcasters in the European Broadcasting Area. While a country can have more than one national public service broadcaster, and therefore more than one member of the EBU, the rule seems to be one Eurovision entry per country, so in countries with more than one Eurovision member they either have to co-operate to select and submit an entry, or agree among themselves which of the broadcasters will do it.
Where a country has more than one member of the EBU, this is typically because they have different public service broadcasters using different languages (e.g. Belgium, Ireland, Switzerland) or different members representing publicly-owned and commercial broadcasters (e.g. UK) or - less common nowadays - different broadcasters for radio and television (Tunisia, Armenia). I’m not aware of any country that has separate public service broadcasters for sub-national divisions like the Faroes, etc that are members of the EBU. I suppose if that happened, then a push might be made to challenge the one-song-per-country rule so as to admit songs representing sub-national areas. But, so far, it hasn’t happened.
Under current rules, no, because they have no broadcaster who is a member of the jEuropean Broadcasting Union. The Faroese have applied for membership but were rejected back in 2010, on grounds that only independent nations can be members. More recently, the rules seem to have relaxed a little (Scotland and Wales had separate entries in the Eurovision Choir competition, e.g.) and the Faroese have talked about reapplying.
I would think Gibraltar wouldn’t be interested anyway on the grounds that it would suggest that they were Not Part of Britain, which they very much insist they are. And they certainly wouldn’t want to cast any doubt on the matter in a venue that included Spain, who might get a little overly excited about it.
Population-wise they’re on par with San Marino so there wouldn’t be an issue on that basis.