The term “Flat Earthers” is usually used in a metaphorical sense, an analogy for people who believe similarly discredited, but more widely supported things such as Young Earth Creationism, vaccine-autism links or that humans are not affecting the climate. However, even in the age of space travel and widespread use of round the world trips, some people still couldn't accept the Earth is curved.
A genuine Flat Earth Society ran from 1956 to 2001, peaking at 3,500 members, keeping up a stream of newsletters during that time. Their dedication appears to have been based on an interpretation of certain Bible verses, and an extreme version of the belief that everything in the Bible was literally true, including the bits about the sun being told to stand still.
More recently, the society has been “refounded”, including a substantial website and wiki. As one Vice reporter found, they're happy to respond to any arguments for the roundness of the planet. What is less clear is whether the modern day members really do believe in the flatness of the Earth, or if they're just having a joke at everyone's expense, journalists included.
If they're hoaxing they clearly have no intention of letting on, but there are a few things to arouse suspicion. For one, their website seems remarkably technologically sophisticated for people who reject the whole of modern science. They also sell a lot of merchandise, which might provide a reason to keep up the charade. Finally, take a look at some of the claims made on the site. References to "competing Flat Earth models", and ice walls “40-50,000 feet high” that apparently no one has seen might be more indicative of some science fans having a laugh than Biblical literalists who tend to be pretty certain about everything.
However, anyone who has tried debating other forms of science deniers will know that sometimes it can be hard to tell satire from delusion. There seems little doubt, for example, that there really are people who maintain the sun goes round the Earth or see Einstein's work as a liberal plot. Indeed some of these sort of ideas keep worming their way into “mainstream” Global Warming denial.
So is the modern Flat Earth Society for real? You be the judge. “It is also possible that NASA does not actually know that the earth is flat and since its inception has simply been faking the concept of space travel, never bothering (or unable) to go any farther than the edge of the atmosphere. The earth is portrayed as round in NASA media because the general public already believes that it is round.”