Apparently there are people who really, honestly think that the Earth is a flat disc, and that all we know about astronomy is just a bunch of lies (perpetuated by a gigantic, world-wide conspiracy.) And I don't mean just a few lunatics rambling in their basements, but a relatively substantial amount of people who take it seriously and actually try to rationalize it. They have websites, forums, books and "documentary" films on the subject.
The funny thing about it is that when you read their web pages and online forums, and watch their videos, it's really, really hard to tell if they are being serious or if it's just a parody. (Poe's law is in full effect here.) However, apparently at least some of those people are really being serious about it.
What's even funnier is seeing how they have to struggle to argue their position. As more and more undeniable evidence has come forth during history, they have to keep changing their arguments.
In the distant past flat earthism was much simpler to believe: It was believed by many that the Earth was just a flat disc and that the Sun and the Moon just orbited around it, so that half the time they would be above the disc, and half the time below it. This was believable to them because they didn't have experience on the vast size of the Earth.
However, this model poses a serious problem: It's undeniable that day and night do not happen at the same time on the entirety of Earth. At some parts there's day while at some other parts there's night, at the same time. This is such an utterly undeniable fact that basically no flat earther even tries to deny it. Therefore they have had to come up with something else to explain it.
They thus invented the concept that the Sun and the Moon are in fact spotlights that are floating above the Earth in circular paths, at a relatively low altitude of a few tens of kilometers. (AFAIK they don't even attempt to explain how exactly this works. Basically it works by magic. And of course the hundreds of thousands of astrophysicists around the world are all in a huge conspiracy to keep this secret.)
Of course that still doesn't work: If this were indeed the scenario, then a "sunset" would consist of the Sun getting smaller and dimmer as it moves away from the observer until it effectively "turns off" (when the observer gets out of the spotlight's penumbra.) Instead, what we have is an unchanging Sun that clearly goes down and descends beyond the horizon (from our perspective.) The Sun doesn't "turn off", it goes below the horizon, quite clearly and visibly. The "spotlight that's hovering above a flat Earth in a circular motion" simply can't explain this. Yet the flat earthers insist that it does, and stubborningly deny the plain geometric contradiction between this alleged system and what we observe, no matter how clearly you try to explain it to them.
Over a hundred years ago it was also easier to believe that nobody had actually visited the Antarctica and especially the south pole. Only a few people had allegedly visited there, and they were just claims that some people had made and that some newspapers had published... They could all just be unfounded rumors or outright lies (from people who wanted to get famous by alleging having done something they didn't actually do.)
However, in the modern world it's harder and harder to argue that the south pole is unreachable, and that what's known as Antarctica is actually a gigantic wall of ice at the edge of the Earth (as many flat earthers like to claim.) Thousands and thousands of everyday people have visited Antarctica and even the south pole, the Antarctic continent is an active site of research for many independent countries, lots and lots of independent documentary makers have made television documentaries about Antarctica, and so on and so forth. The amount of everyday people who have been there is just enormous. As time passes, it's exceedingly implausible to keep claiming that there's a huge conspiracy to keep people from going to Antarctica.
Of course that doesn't stop the flat earthers from making such claims even today. They do.
The handwaving gets pretty wild when we start talking about distances. For example a plane flying from South Africa to the southern parts of South America takes significantly less time than what the flat Earth model would suggest. Flat earthers have a hard time explaining this away. (The conspiracy theories usually get pretty wild at this point.)
Of course the most damning phenomenon with respect to the flat Earth hypothesis is the position and motion of stars. Whenever you are on the northern hemisphere, you will see on the sky the North Star in the constellation of Ursa Minor, and if you observe it for a day you will see that the constellation and all stars rotate around it. Moreover, if you eg. sail from northern Europe to northern America, you will constantly see it.
Likewise wherever you are in the southern hemisphere, when you look away from the north, you will see the Southern Cross, and if you observe it for a day, you will see how it and all stars rotate around it. If you sail eg. from Australia to South America, you can see it all the way through. Moreover, if you sail from the northern hemisphere towards the southern, you will see how the North Star descends beyond the horizon as you approach the equator, and then how the Southern Cross raises from beyond the horizon from the opposite direction.
All this is trivially explainable if the Earth is a rotating sphere with stars surrounding it, but a geometrical impossibility if the Earth is a flat disc.
There's a variant of the typical flat earther that actually doesn't try to deny that the Sun looks like it's descending below the horizon, that ships that sail away from the shore look like they descend below the horizon, or even that we have achieved spaceflight and have photographed the Earth from space. Instead, they try to conjure General Relativity into the mix to explain all this.
You see, according to them, General Relativity predicts that light bends near massive objects. Therefore when the light reflects from a receding seaship, it bends down. Therefore from the shore it looks like the sea and the ship with it is going down, which is why it looks like it's descending behind the horizon. The same with sunsets. Also the photographs taken from space are really photographing a flat earth; it's just that light bending makes it look like a sphere instead.
The major problem with this pseudoscientific explanation is not that the numbers are way off (in order for light to bend that much, the density of Earth would have to approach that of a neutron star.) No, that's not the major problem. The major problem is that they have got the effects of the light bending backwards.
If light coming from the seaship (and the sea itself) bends down, when looking at it from the shore it would look like the sea is actually curving up (and thus that the ship is climbing upwards.) From the shore it would look like you are on the bottom of a bowl, with all the landscape around you curving up. Likewise when photographed from space, it would look like you were photographing the inside of a bowl. It would not look like you were photographing a sphere. It's just hilarious how utterly wrong they understand this.
In short, reading about how flat earthers struggle to explain away all the facts is quite entertaining.