NYT wrote an article on this recently

"Appealing though it may be, the myth of a ship’s captain presiding over the nuptials of dewy-eyed couples has for most of the last century been pretty much just that. And yet the demand for weddings at sea has grown to the point that some cruise lines, operating under foreign flags and laws, have found ways to perform legal unions in international waters with the ship’s captain as officiant."

It may be well known that only priests, judges and captains can marry people, but it's not actually true, at least in the UK. I was (very) surprised to find that I'm allowed to do it myself. As my church is not Church of England or Quaker, the couple would have to obtain a marriage licence from the Register Office (basically notify them in enough time for a check that they can legally marry) and there's some differences in record keeping. However from then on it's governed by whether the church will recognise me for the purpose, which it does (I'm a lay preacher). The same seems to apply to many other religious organisation other than the two above, or the Register Office itself. It does not apply to at least some non-religious organisations, e.g. the Humanist Association (other than in Scotland - remember that's a different country with a different legal system from England), so I'm not sure where that leaves ship's captains.

BTW, no, I'm not going to do one!