The "AIRES" project is apparently misspelled on the visible screens during the scientific briefing. If, in fact, it refers to the astrological sign or constellation, it would be spelled "Aries." If it refers to the god of war, it should be spelled "Ares."
In the hospital Woody Allen's character has on his doctors hat on wrong purposefully. However, in the chase scenes while fleeing and outside the hospital it changes and is on correctly like Diane Keaton's character.
The Volkswagen could not possibly start after 200 years in a cave. Gasoline typically has a 30 day shelf life. After 30 days the aromatic parts of the gasoline noticeably start to evaporate, and it begins a chemical breakdown process into gum, resin and varnish which cause misfires and poor starts, poor performance, and could cause engine damage. Degraded fuel in a running engine also leaves deposits in the carburetor plugging essential passages and can also leave deposits on the intake valve stem causing it to hang up and not close properly or even not to close at all.
The Woody Allen character, under hypnosis, states that his first wife wanted a divorce in 1963 because he "drank his waterbed." This could have never happened. Waterbeds had not been patented until 1971.
During the presentation about the Leader it was said that none of the doctors know each other for reasons of security yet before this presentation one of the doctors calls Milo by a specific name clearly indicating they know each other.
The goof item below may give away important plot points.
The whole point of reviving Miles is to have someone without any record in order to infiltrate the Aries Project, but Luna is able to do so just as easily despite the fact that she is a known fugitive.