Factual error: The German agent pretending to be an American pilot is wearing a one piece flying suit from the 1960's when he is questioned by Hogan and the British escapee in the barracks. It even has the US Air Force logo on the sleeve, which didn't come into use until 1947, after the war.
Factual error: At the end of the episode, Major Nelson leaves his Cocoa Beach, Florida, house and drives down the street. Behind the house at the end of the street (which is the home used for the Stephens family on "Bewitched"), there is a mountain. There are no mountains in Cocoa Beach.
Factual error: We see how the accident-prone Wilton Parmenter was a soldier at Appotomatox during the Civil War when he sneezed. Union soldiers mistook his sneezing for a call to regroup and charge, winning the battle. Wilton Parmenter was awarded the Medal Of Honour. Accident-prone, even in his moment of glory, when the medal was pinned on Wilton's chest it pierced his skin, drawing blood. Wilton was therefore awarded the Purple Heart, becoming "the only soldier in history to get a medal for getting a medal." This was not possible, for the "Purple Heart" (or "Badge Of Military Merit", to use its proper title) awarded to US soldiers wounded in action, was only instituted in 1932. (Before somebody corrects me, George Washington did institute the award for some distinguished combatants during the War Of Independence, but it was never awarded during the nineteenth century, and revived in 1932).
Factual error: The Robinsons exceeded the speed of light due to impurities in chemical rocket fuel. No chemical rocket fuel comes remotely close to the energy content needed to even approach the speed of light. That would require exotic fuels like antimatter. By Einstein's theory of Special Relativity it is impossible to exceed the speed of light with any form of rocket propulsion. Faster than light travel and time travel might be possible with the warped spacetime of Einstein's general theory of relativity and phenomena such as wormholes. For plausibility the opening scene should have been an encounter with a wormhole.
Factual error: If there really was a hatch leading into mid-air in the airplane bathroom, instead of being soundless like shown, it would be incredibly loud and Max would have noticed. Also, the sudden pressure change would have sucked him out no matter what he did to stop it.
Factual error: It is quite true that atoms that do not occur naturally are bad for humans. However, in a space age as this story's, that would apply to most or even all planets. It would be a well known risk that didn't require an abnormal specialist to diagnose. (This fact rarely figures in other SF, because it would severely limit most stories).