A photo shows Thornton's character wearing a pressure suit and standing by an X-15 rocket plane, implying he flew it while in the Air Force. The X-15s final flight was in October, 1968... meaning the character would be well over 60 years old.
Farmer is able to communicate with his ground controller, Sheppard, throughout the flight, except when his radio system is not operational due to a malfunction. In fact, radio transmissions are only possible if the antenna of the transmitter has a "line of sight" to the receiver. Farmer could not speak with his ground controller when he was on the other side of the Earth. NASA uses a series of antennas located around the world to receive and relay radio signals to Houston, the home of NASA Mission Comtrol at the Johnson Space Center.
The type of Atlas rocket show in the film was designed without a rigid internal skeleton. It was held up by pressurized fuel tanks. If the capsule were loaded atop as shown without the rocket being fueled and pressurized, it would have collapsed under its own weight.
For his first launch, Farmer fuels the rocket with kerosene and hydrazine. Either one can be used as a rocket fuel, but both are worthless without an oxidizer. Even combined, he never would have gotten an ignition, let alone a failed launch.
Farmer's orbital path, due to the rotation of the earth, would not pass over the launch point until 24 hours or roughly 18 orbits have been completed. After nine orbits the path of the capsule would take it mostly over Europe and Asia, then dipping into the Southern Hemisphere, nowhere near the southern U.S.
The producers might originally have envisioned a 24-hour flight (and filmed it that way) but for some reason the number of completed orbits was changed to 9 in the script.
A spacesuit requires external connections to a) supply air for breathing and cooling and b) remove carbon dioxide and excess heat. Thornton's character is shown wearing a spacesuit in many situations that would have resulted in both overheating (when he wasn't wearing the helmet), and suffocation (while wearing the helmet closed), since he had no portable air conditioning unit or other umbilicals connected.
During the flight, Farmer's capsule comes very close to what apparently is a communications satellite. Typically communications satellites are in geosynchronous orbit 22,236 miles above the surface of the earth. Since Farmer's orbit was planned for 100 statute miles perigee (and presumably an apogee around 175 statute miles, typical for Project Mercury), there is no way he would ever come within sight of a communications satellite.
Upon re-entry, rockets are fired for the purpose of slowing down the capsule to sub-orbital speed, but the capsule is still traveling many thousands of miles per hour relative to the ground below. However, when Farmer fires his retrograde rockets, his capsule seemed to simply come to a stop, and drop vertically to Earth. In addition, he fired his rockets after appearing to fly over Baja California, for a landing in Texas, while in reality, descending manned spacecraft fire their rockets thousands of miles in advance of their desired landing spot.
In the interview during the credits, Jay Leno states that Mr. Farmer completed nine orbits of the earth. At about 89 minutes per orbit (typical for Project Mercury flights) his total flight time would be around 14 hours (13.5 hours in orbit plus 30 minutes entry/reentry). If he launched at 9 am, his return to earth would have been around 11 pm that same day. Yet the movie has him returning to earth sometime after sunrise the following day.
When the rocket launches, incredibly, the wooden barn remains unburned. In a scene during the second launch, exhaust smoke can be seen coming out the bottom of the barn walls, so it's safe to assume Farmer has designed his barn to overcome the explosive nature of the launch.
During the closing credits, an additional scene is shown where the character of Charlie Farmer appears on the Tonight Show with Jay Leno. When Jay greets Mr. Farmer before Charlie sits down, Jay can be overheard saying what sounds like "How are you, Bob?", if fact, he said "How are you, pal?" This is also indicated by the subtitles.
While Farmer and his wife are in bed at night she realizes his wedding ring is missing. He assumes it is in the first capsule that he had built at that time and later crashes. In the second capsule he builds later while in space he has a power failure that he later finds the wedding ring to be the cause (which he just finally just found).
When the first, failed launch attempt causes the rocket to tear across the terrain horizontally, the rocket shoots right through a billboard, leaving a hole only about three feet across, much smaller than the diameter of the rocket.
Carnival rides use electric motors connected to huge generators to turn them. When the Farmers have the carnival ride on their property we see a large overhead view of the carnival ride but there is no generator or electrical cord anywhere. The cord is obviously buried and the generator is behind the camera.