Factual error: Marcus is spun off and he's "on course to intercept the sun in about a week." The sun is 94,000,000 miles from earth, even assuming 2 weeks he'd have to be travelling at about 280,000mph. Orbital speed is about 17,000mph, no way he was flung away from the satellite 16 times faster than that from what we saw.
Mark and Mallory Go to Washington - S1-E3
Continuity mistake: The digital clock behind the head of the Air Force doesn't match the time elapsed between shots - we see him with it at 14:00:54, 10 seconds pass of continuous conversation, then it's at 14:01:14 when he says "you seem pretty calm."
Space Flag - S1-E5
Other mistake: The kill counter on the screen displays the wrong counts... should display "Space Force 14", "Air Force 15". (00:26:08)
Continuity mistake: The digital clock behind Naird skips backwards - it's at 19:34:54 when he says "I'm so sorry Adrian", but the next time we see it it's just ticking onto 19:35:01 when several more seconds have passed.
It's Good to Be Back on the Moon - S1-E9
Factual error: The Apollo 11 flag gets knocked over by the Chinese rover...but the flag was blown over by the Eagle's takeoff thrust, witnessed by Buzz Aldrin. This was confirmed years later by the Lunar Reconnaissance Orbiter, which showed shadows of the flags planted at all six Apollo landing sites, bar Apollo 11's. The other 5 were all planted further from their landing craft precisely for that reason. It's also widely assumed that the flags on the moon would have been bleached white by decades of unfiltered sunlight, not in good condition like the one shown.
