In the episode that discusses the flight of Apollo 7, there is a scene in which the crew is descending from the tower elevator. In the background, a Saturn V rocket is clearly seen, but Apollo 7 was powered by a Saturn 1-B rocket.
In the first episode when you see Al Shephard's Mercury capsule climbing away from Cape Canaveral it is obviously a contemporary satellite image as both of the pads from Complex 39 are visible, as is the Vehicle Assembly Building and the Space Shuttle runway. Complex 39 from which the Apollo missions were launched only commenced construction by the Army Corp of Engineers after 1961.
In several episodes, all set in the 1960's, Western Electric telephone handsets with modular cords are seen. Modular connectors were patented in 1973 and did not appear on Bell System equipment until 1975.
In part 6, "Mare Tranquilitatis", when Armstrong and Aldrin are exiting the LEM simulator after crashing there is a sign next to the door reading "Remove shoes before entering cabin" but we hear the sound of hard shoes on the catwalk.
In episode 10, when Dave Scott is demonstrating the famous Galileo "feather vs hammer" experiment, he is seen holding the hammer the right way up. The archive footage cut into this shows the hammer wrong way up (so that the head wouldn't catch his glove when dropped). In the final shot, the hammer is right way up again and hits the surface shaft first.
In Part 10, "Galileo Was Right", Astronauts Scott and Irwin were attempting to pull the drill out from the lunar surface. The drill was shown to be at the height of the astronauts' chests. When the shot returns to mission control, archived footage shows the drill to be at the height of the astronauts' heads.
When Armstrong is training in the LEM landing simulator at Ellington Field near Houston Texas, mountains are visible in the background in one scene. Ellington Field is at 32 feet MSL (Mean Sea Level) and there are no mountains or even large hills in that area.
In Episode 3, Mark Hamon as Wally Schirra states that Gordon Cooper left the program when he is talking to Deke Slayton on the beach. This "conversation" would haven taken place in 1968. Cooper did not leave NASA until 1970, when Shepard and Slayton passed him over for command of Apollo 13.
In part 6 "Mare Tranquilitatis" during the landing sequence, there was no radio transmission delay between the capsule and Houston. For example, when Armstrong asked Houston a question the response came back almost immediately. A round-trip radio signal across the 240,000-mile distance would have a lag of close to 3 seconds. This was even mentioned by Gene Krantz earlier in the program.
In Part 6, Mare Tranquilitatis, the GUIDO flight controller's computer screen shows "Program Alarm 1202" during a particularly dramatic moment in the landing sequence. The same screen contents are also used earlier in the episode, during a simulation run, where there is no such program alarm.
On several occasions we see the descent engine of the lunar module blowing a bright white flame like a Star Wars spaceship. In actuality, the hypergolic fuels used in the LM did not produce a visible plume. In the vaccuum of space, the only visible effect would be the dust being blown away or, in the case of the ascent engine, the descent stage being blown apart.
When Neil Armstrong and Buzz Aldrin walk on the moon, Aldrin says, "Magnificent desolation," and Armstrong says, "Magnificent sight out here." These lines are historically accurate, but they were actually said in the opposite order.
In episode 6, the clip showing Buzz Aldrin descending the ladder was obviously taken from episode 12. This is evident because a Lunar rover can be seen stowed on the LM, a feature that Apollo 11 did not have. Also, Neil Armstrong was not near the ladder of the LM, taking pictures of Aldrin, as in the real mission.
In the "1968" chapter, we see a TV reporter broadcasting with a massive picture-window view of the Saturn 5, and then feeling the shock wave from engine ignition just before the window falls in. The view is from 1/4 mile SE of pad 39A, a dangerously close point for anyone. The press site is 3.1 miles west of the pad. Also, the incident happened to Walter Cronkite at the Apollo 4 launch, Nov. 9, 1967. The launch depicted here (coinciding with the King assassination) is Apollo 6 on Apr. 4, 1968.
During the Apollo 11 landing in "Mare Tranquilitatis", flight director Gene Kranz repeatedly includes "booster" in his controller polls. Booster controllers were present only during powered flight on the Saturn booster. Their job was done at the end of translunar injection and they were long gone by landing.
As the Apollo 8 spacecraft travels in orbit around the moon, one shot shows the moon's surface visible through a semi-transparent antenna. The 3D rendering of the antenna did not have the opacity set high enough, resulting in this effect.
In episode 5, "Spider", when Astronaut Rusty Schweickart is shown entering the LM for the first time while in space, the shot has been reversed. The NASA patch on his space suit is shown in reverse and on the wrong side.
In Part 9, "For Miles and Miles", at the beginning when Alan Shepard is brought to his wildcat oil rig, and he gets dizzy and sick, they arrive in a 1965 Ford Thunderbird. The next scene shows him being examined because of the dizzy spells he was having and the flight surgeons report starts off by saying the date is June 10, 1963, far too early for even a 1964 T-Bird to have been on the road.
In one scene showing the earth from space, the famous Apollo 17 photograph is used where North Africa and the Arabian peninsula are visible, but the photograph is backwards in a reverse image. The negative was flipped before printing.