Technical details. (spoilers)
I am enjoying this series, but there are some style over substance problems unrelated to the gender issues that have some people's panties in a twist.
1. Using a Space Shuttle as a ferry between the Earth and the Moon. This is just silly. The shuttle has lots of structure (wings etc) that is very massive and useless in lunar orbit. All this extra mass requires lots more fuel to get out of Earth orbit, reach the moon, enter lunar orbit, leave lunar orbit, then enter Earth orbit prior to entering the atmosphere to land.
The external fuel tank and solid boosters have almost enough fuel to reach Earth orbit, then the much smaller OMS engines take over from there. A large orbiting refueling facility would be required to be maintained in Earth orbit to allow the shuttle to act as a ferry between the Earth and the moon. The SSME's used to leave the launch pad lack restart capability and require cryogenic fuel. The OMS engines use a fuel that is less efficient than the liquid hydrogen/LOX used by the SSME's. Filling the payload bay with fuel will provide enough fuel to go to and from the moon, but the added mass is too much for the SSME's and SRB's to lift the shuttle into Earth orbit in the first place.
It would be much more practical and believable to use something like a larger Apollo type Command Module and Service Module as a ferry between the Earth and the Moon. A booster with an F-2 engine can be used for TLI like on Apollo then expended or returned to Earth orbit for reuse. A cheap return vehicle like an Apollo CM can be used to land back on Earth.
2. Little visible infrastructure at the new Jamestown base. We never see the lunar return vehicle on the lunar surface. No one is ever seen plugging in the rover between uses later on.
3. Apollo 15 changes its mission too much for credibility. Changing to a southern landing zone near Shackleton then using the rover as a winch for descending into the crater? How did they get the wire mesh wheel off of the rim? Hammer?
Why did they bring 100 meters of cable to the moon? That is heavy. They just happen to have it for something else not mentioned in the episode? Ten meters of cable for various experiments is more likely.
4. Exhaling cigarette smoke into a vacuum hose is not going to prevent Tracy from stinking up and polluting the Jamestown atmosphere. Perhaps if she was smoking inside of a helmet with the vacuum hose sucking air from it then exhausting into an activated charcoal filter?
5. Cobb and Wubbo would have had symptoms (vomiting, diarrhea, fatigue) of radiation sickness within several hours of their high rate of exposure during the solar storm. Wubbo's dosimeter read 200 rem. Although this causes "mild" radiation sickness, it can be fatal without medical care. There is no way Cobb could have hidden this from the Jamestown crew.
6. Jamestown interior scenes show normal Earth gravity. Only in the lunar surface exterior scenes do we see 1/6th gravity.
7. The rifles the crew carries in the season 2 episodes seem to just be M-16's with small scopes and a white coating of paint. Better that they they used a CAR-15 with a 14" barrel or the new (back then) M-4. A scope with a much wider eyepiece would make aiming the rifle while wearing a bulky spacesuit much easier. The polymer stock and handguard would also need to be replaced with metal as a coating of paint might not be good enough to reflect enough radiation to prevent softening the material.
8. Pathfinder phoenix missile. The version they refer to has fins for steering in the atmosphere and no thrusters for operation in space. Fins are useless outside the atmosphere.
9. There is no way to hook up and refuel a service module near the moon, especially in the ten minutes they had left prior to reaching the point of no return during the Apollo 24 mishap.
I think the writers were a bit lazy. A bit more effort would make this a better show to watch.