Factual error: A 250-year-old sea chest recovered from the ocean floor is opened, and all its paper items are perfectly dry: the log book can be opened and read, and the letters can easily be unrolled. Highly unlikely. The chest's wood and pitch would deteriorate (salt water is extremely corrosive), and the paper should be damp and crumbly, at the very least. Even more likely, it would have become a lump of unreadable sludge.
Audio problem: At night when MacGyver and Jack are sneaking past the guards behind a mule, we see about 8 to 10 light bulbs hanging overhead. MacGyver later pulls the cord down and all the lights begin to pop. There are about 16 pops, many more than there were bulbs on the cord. (00:24:50)
Factual error: Some factual errors regarding Basques: 1) Basques have not been fighting for ages against France and Spain, and they do not kidnap US citizens. 2) Basques do not look, in any way, like South Americans. 3) Violent Basques are not guys "up on a mountain" with old rifles, but terrorists with modern weapons.
Continuity mistake: In the scene where the gang are chasing after the two kids, it is raining and the street is wet. They stop chasing and turn around when they see a police car go by. In the first shot of the police car, the street is completely dry. (00:05:45)
Visible crew/equipment: As Amy and MacGyver are chased around a nuclear reactor just before the bad guy gets nuclear steam on him, you can see a pair of legs in a top row window around where they are walking. Remember the nuclear reactor was hot with radiation so no-one would be wandering around.