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.
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.