In this first episode there are communications (phone calls, of a sort) between people on the moon, and people on earth. They are depicted as completely, and instantly conversational (just like a normal phone conversation on earth). In reality, the distance between the earth and the moon (and the limitation of light speed), means that when one person finishes talking, it will be approximately 2.5 seconds before the other party will begin to hear their response (and that is if the other person starts to talk, the instant that the other party finishes). Obviously, those delays increase, the further one travels from earth, such that eventually, conversational communication becomes completely impractical.
Before leaving the moon, the group is setting up a live video link on the moon surface. The video camera is mounted on a regular 'earth' tripod. All other equipment however is specially designed for use on the moon and in space. A normal tripod would never be used.
A fire breaks out in the ship on the way from Earth to the Moon, when a floating liquid is ignited by the Commander's T-shirt. The globular flames rapidly rise up the shaft where they are captured in an upturned container. But they are in zero-G, so there is no convection; heat does not rise, the burning liquid would stay where it is.
When the rocket is on the Moon before launch, the "steam" from the cryogenic fuel/oxidant is shown flowing downward. This would not happen in a vacuum such as the Moon.
The journey from the moon to Mars (about 39.7 million miles) is said to take 8 months. In fact, a spacecraft traveling 17,500 MPH (as noted in Negative Return (2020) ) would only take a little over 3 months to travel that far.
During the fire in the conduit scene a rack mounted server is shown embedded in the wall with HPE G5 style hard drive cradles clearly visible. These were used circa 2004 to 2012 and have since been retired from service. Obviously these were used a low-budget prop.
The crew are show distrust in the commander, and NASA considers removing her from the crew 24 hours before mission start. Astronauts go through extensive crew training and psychological evaluations prior to going into space to prevent exactly this sort of thing. This is unrealistic given the weight of their mission and done to add drama.