At one point Danielle Poole throws a metal box to Ed Baldwin, and it doesn't drop at all. The Moon's gravity is lower than Earth's, but it's not nonexistent, so the box should have dropped somewhat lower.
When Gordo and Dani return to the Command Module to return to Earth, Ed accompanies them when the leave the moon using the LEM. The image shows the entire LEM. This is wrong. When The LEM blasts off from the Moon, it leaves the lower portion behind. Thus Ed can't return to Jamestown Base.
The American astronauts on the moon are constantly communicating data using the metric system - NASA has only used the metric system since 1990 in our timeline. The entire series takes place in an alternate timeline, and NASA adopting the metric system much earlier than in our timeline is one of the differences.
Throughout the episode it's a major plot point that the astronauts can't relay the situation at Jamestown Base to Houston out of fear that the Soviets may be listening. But later they're casually discussing the state of Gordo during EVA at the mining site. The space suits' radio traffic isn't encrypted and the Soviets certainly listened to the conversation.