It would be interesting to try to estimate the percentage of the earths population which would have to be lying about a flat earth.
As I wrote somewhere here a month or so ago, we (me an my colleagues) did try such exercise on a lazy friday eve.
I admit, we weren't exactly sober, but we estimated some 40% population of our country (Ukraine) to have to keep quiet since mid 50s. This includes all sorts of astronomers, geodesists, rocket designers and builders, politicians and local authorities obviously, accountants in all related government institutions... And lots and lots of other people of various specialities indirectly involved in building the rockets and satellites for USSR/Russia, e.g. people on metalwork factories making rocket/satellite parts, tracking stations, observatories...
Not sure how it extrapolates to entire Earth, but I imagine it's still a huge chunk of Earth population. If there was a hoax making all these people's jobs impossible, it would've slipped out over all this time.
I would guess a much lower percentage, although it still would be quite high. For example local authorities or the people making rocket/satellite parts do not necessarily know that what they are working on is a fraud.
On the other hand, every pilot and flight director, most members of any naval operation and everyone who has ever been to antarctica and seen a midnight sun/midday night would have to be included. (If you want to be nitpicking you can include every person (equipped with a bit common sense) in the southern hemisphere, since the motion of the stars there can not be explained under FET, but then you would have to include everyone who has ever seen a sunset, and that would make lots and lots of people. But I guess those can not be called liars, since even most FE members would have seen a sunset.)
Also, it is not just NASA, also Roscosmos (and ESA and so on). After the fall of the sovjet union, where there was a lot of anarchy and whole weapon arsenals supposeldy vanished, this conspiracy stayed intact, which says a lot about how well-organized it is. Why the russians participated in a conspiracy which eventually ruined them is still a mystery though.
Edit: Oh right, I forgot: We have google earth, which maps our planet as a globe with quite accurate precision (say we allow a 5% error in measured distances in google earth). Then the earth can not be flat, since curvature is invariant under isometries (saying if we map something and preserve distances, then we also preserve curvature). So (especially in the southern hemisphere) the distances have to be very different from the ones provided by google earth. Noone noticed until now, so many people in the southern hemisphere have to be included in the conspiracy.