Purpose in the Machine - S3-E2
Factual error: When examining the sand, Bobbi states "According to carbon dating..." Fitz finishes by saying "It's a billion years older than the earth." However, carbon dating is only good at determining the age of something to about 60,000 years old. The only thing carbon dating could tell about the sand is that it is 60,000 years old or older. It certainly couldn't determine that the sand was 5 billion years old.
Purpose in the Machine - S3-E2
Continuity mistake: When May is sitting on the couch talking to her father, the candles behind them are not lit for a couple of shots. They are lit when they start talking and when they finish.
Character mistake: Lance and May walk into a bar in Massachusetts. As they enter, the camera pans and shows a large banner on the front window. The banner is advertising "game day" food and drinks. In very large letters the banner shows Boston Red Socks. The actual correct spelling is Boston Red Sox. (00:05:45)
Chaos Theory - S3-E7
Revealing mistake: In the shot where Lash hauls Rosalind over the parapet to later drop her, you can see that he is actually lifting a mannequin as its legs very fluidly bend backwards while it is being dragged over. (00:32:00)
Continuity mistake: When the Watchdogs destroy the government facility near the start of the episode, 6 shots of Nitramene are fired onto the building's exterior before it implodes. But moments later when Coulson calls Mack, he (Coulson) is watching the news report of the event, and the 6 shots of Nitramene are nowhere to be seen.
Other mistake: Agents say they have seen seismic activity in South Dakota, but from earlier episodes, Daisy and Lincoln visit South Australia to find out Fire Guy.
Continuity mistake: In the opening of episode 20, Coulson meets General Talbot in a saloon that is a secret entrance to their hidden base, a base that is many stories underground. Yet in episode 22 Coulson uses his robot arm to call another jet; it appears outside a window! He can't have a window in his office, if the base is several stories underground. And to verify it isn't a "video window" or "projection screen", Mac throws a chair through it, essentially saying "We are above ground, let's go get in that jet in front of us." (00:19:40)
Chosen answer: Coulson believes he was resuscitated then sent to Tahiti to recuperate. "A magical place," he calls it. But in the "Agents of S.H.I.E.L.D." episode "The Magical Place", Coulson, through a mind-reading machine, learns that he was treated by S.H.I.E.L.D. and a fake memory of Tahiti was placed as a "mask" of sorts over the painful memory of the operation, which involved a drug seemingly of alien origin to repair the damage, coupled with a lot of morally dubious surgery.