There are no totally 'closed' countries in the world for British passport holders. But there are some which are very difficult to visit.
Some countries won't allow nationals of other countries to visit. For example, Armenians aren't generally allowed into Azerbaijan (a few exceptions are made). Israelis aren't allowed into Iran, Lebanon, Algeria and a whole load of other countries under any circumstances. Also, people with Israeli passport stamps or evidence of having been to Israel are not allowed into certain countries.
Some countries make it very difficult for British passport holders to visit unless they are on a tour (North Korea, Iran, Turkmenistan) or have a work permit/support of their employer/are on a pilgrimage (Saudi Arabia). It is almost impossible to visit Saudi Arabia as a tourist, but quite easy as a businessperson with a local sponsor.
There are also some other countries which make it exceptionally difficult for tourists to get visas for. The hardest are often considered to be Equatorial Guinea and Angola, both African oil states- it is extremely bureaucratic and expensive to get vias for these places as a tourist. Yet, it's relatively easy to go if you are in the oil industry and sponsored by you company. Algeria is also very tough.
Other countries are difficult to get visas for because they don't have an embassy in the UK- mostlyFrench speaking African countries. So, even if the visa requirements are quite cheap/easy, the logistics of getting the visa canbe difficult.
Some people mentioned Moldova- you can fly on budget airlines from Luton or Stansted and don't need a visa. Moldova is in no way closed (though few people visit).
Bear in mind that it is generally more difficult for a citizen of any of the countries named above to visit the UK than for a British person to visit that country. People often complain about how 'hard' it is to get a Russian tourist visa, for example. It's not that difficult, you just pay your money and fill in a (long) application. It is MUCH harder for a Russian to get a visa to visit the UK- they have to apply in person at the embassy/consulate. For someone living on the Pacific coast, that means an 8-10 hour flight to Moscow or a 6 day train journey. Fees are about twice that of a Russian visa for a British person (and local salaries much lower) and are non-refundable even if the visa is refused.
Generally, British citizens have it very lucky.