Speaking of Yuri Gegarin's orbital flight, the TV announces the flight time in UTC ("Coordinated Universal Time", with French initials). However, UTC had barely been invented in 1961, and was not yet called UTC.
It would seem that most all of the scenes involving vehicles were shot at one time with the same assortment of cars -- despite location -- appearing throughout the film, including a bright red Metropolitan sedan. Because these are all doubtless collector pieces leased for the film, all are in perfect, spotless condition, regardless of age. Further, a number -- including a blue '57 Thunderbird -- do not change position in the NASA parking lot despite shots being days, weeks, or months apart.
The phone cord used by John Glenn on Pad 14 in Cape Kennedy is one that is metal reinforced to deter vandalism which would not have been deployed in 1961.
The Go/No Go calculation that features so prominently in the film takes place during the descent of Friendship 7, just before parachute deployment. It actually took place long before, when the retro-rockets were fired.
Shots of Cape Canaveral/Kennedy Space Center show the crawler tracks to Launch Complex 39. LC 39 was constructed for launch of the Saturn V rocket that sent the Apollo spacecraft to the moon. Construction on LC 39 didn't begin until 1962 whereas the Shepard, Grissom, and Glenn launches were in 1961 and early 1962. The crawler path would not have been created yet. Furthermore, the Vehicle Assembly Building would not have existed yet, either.
In Al Harrison's office, there are two model aircraft on the shelves behind his desk. The one on the left appears to be a C-130, which was in production at the time but did not have the light-gray paint scheme depicted. The one on the right appears to be an unpainted wooden model of a C-5 Galaxy, which was not designed by Lockheed until 1964.
In a scene purporting to take place in 1961, before September when new 1962 car models began selling, the rear of a 1962 Chevy Impala is seen in the NASA parking lot.
When Dorothy Vaughan (Octavia Spencer) is caught using the 7090, there's a shot where one of the men goes over to the printer and looks at the output. The soundtrack is that of a dot-matrix printer, while the printer shown (the slower IBM 716) is a 150lpm typewheel printer, based on the IBM 407 accounting machine, which sounds completely different.
In the meeting at the Pentagon, Katherine is calculating John Glenn's splashdown coordinates on a chalkboard. At one point she's writing a number that has a "530" thousand figure in it, but she starts to miswrite it as "350," then immediately backs up to overwrite it as 530. When she backs away from the board, the figure is correct but doesn't show the overwrite she just made.
In a key scene, Al Harrison, in Mercury Mission Control, sends Sam to find Katherine, in the West Computing Hall, and have her check some calculations. Sam runs to find her, and the two of them run back. West Computing Hall is part of NASA's Langley campus in Virginia. Mercury Mission Control is at the Cape Canaveral launch complex in Florida.
When the ladies' car breaks down, Dorothy Vaughan says that the starter broke. A starter breaking would not cause a car, which has a running engine, to break down. Furthermore, she says she'll just bypass the starter and proceeds to use a screwdriver to short something under the hood and the engine starts. This is impossible, an engine with a bad starter will not start unless the car has a clutch, in which case they would have to push the car to get it moving and then pop the clutch to start the engine. Most cars in the 50s had clutches and this would have been the only viable solution to get the engine started.
During the flights of both Alan Shepard (Freedom 7) and Gus Grissom (Liberty Bell 7) there is a large global tracking map shown. Neither of these flights went more than 200 miles from the cape.
In the trailer, Mahershala Ali's character is referred to as Colonel Jim, and he wears two gold oak leaves on his uniform. The US Army does not have any rank symbolized as two gold oak leaves. A major wears a single gold oak leaf, and a lieutenant colonel wears a single silver oak leaf.
The IBM 7090 was repeatedly referenced as being able to perform 24,000 calculations per second. In fact, the 7090 could do 100,000 floating point operations per second (100KFLOPS).
Johnson performs an off-the-cuff calculation on a blackboard during a briefing, starting with 4 digits of precision; her final results have 7+ digits of precision, something no mathematician in the 60s would have wasted their time on. Such a wasteful, excessive, and overly-precise number is the result of a modern digital calculator, not a human computer.
At the beginning of the movie, we see a Soviet rocket putting the dog Laika into space. We see a Vostok capsule on top of the rocket; Laika actually was launched inside a Sputnik capsule. Vostok capsules were only used for manned flights.
The wind tunnel crew repeatedly refer to the crew compartment skin and various hatches as the heat shield, while in reality the heat shield is actually situated at the bottom of the capsule, beneath the pilot.
The computations on the blackboard would have required at least a slide rule if not a book of trig and log conversions. It is possible to do the calculations using Taylor series conversions, but that would still require a (Friden?) calculator.
When the three women's car has broken down, Dorothy Vaughan (Octavia Spencer) says the problem is that the starter is out. A starter's not working wouldn't cause a car to break down, just make it unable to start. When they leave work later that night, the car starts just fine.
The new mainframe is referred to at one point as the "seven-ninety". IBM referred to the 7090 as the "seven-oh-nine-tee", because it was the transistorized "T" version of the 709.
When Shephard is waiting on the launch pad for lift-off, there is a delay attributed to a problem with liquid oxygen. Mercury I and II used the Redstone rocket, which was solid-fueled and used no liquid fuel of any type.
Mahershala Ali's character is a Lieutenant Colonel in the National Guard which means he has roughly 15-17 years of military service. Despite this his uniform bears only his rank and the crossed cannons of artillery. Missing are unit badges, qualification badges or any commendations including the Combat Infantryman Badge or the Advanced Infantry Badge.