Roundy,
If you prefer there's a lot of other stuff from the 18th, 19th century and early 20th century which doesn't fit the FE model and predates NASA. These include the very detailed surveys of distances by the French Academy of Sciences (incidentally, the fact that gravity varies slightly was already known to them in the 1790s. That's one reason they needed to go do detailed surveying of the Earth, because pendulums behavior changed where you were. This isn't a problem for FE in general but it is a problem for FE versions with universal acceleration.). That work gave very precise measurements for France and Spain which consistent with the spheroid Earth. At about the same time, or slightly before, many of the colonial empires were making very precise measures of their empires, and the navies were using great circle routes and careful measurements of longitude to navigate. In the 19th century, these maps was further mapped out in detail and confirmed using precise observations of the sun, moon and planets. Later in the 19th century, work was done which detected stellar parallax, completely explainable by a moving speroid Earth. In the 19th century, scientists went on to use stellar spectroscopy to estimate the composition and temperature of the sun. In the early 20th century, maps of the Earth and the surface below the Earth were made by seismologists measuring the length of time and strength of different types of earthquakes. That's an ongoing project with literally thousands of stations worldwide and the data easily accessible.
There's so much basic stuff that FE doesn't have any explanations for that the only (marginally) viable explanation is for an early conspiracy, which not only started off early but controls a lot more than just NASA. This reflects a general problem FE has: it doesn't produce interesting new science. One major distinction between good science and bad science is whether or not one gets new ideas that predict more about the world in testable ways. Good science does this. Bad science just keeps leading to more and more ad hoc hypotheses to protect itself from falsification. This is a point that was made by Lakatos, and as philosophers of science go, he's one of the more readable. It seems pretty clear what is happening here: FE is quintessential bad science.