I love the idea of a 60's-set F4 movie in the MCU where they time travel to the modern world eventually. The only question is...would they have been kept secret from the public back then? Because up until "The Avengers", the world of the MCU is very new to the notion of people with advanced powers and abilities. Even Natasha says it to Hawkeye, "This is monsters and magic and nothing we were ever trained for." It would be kind of hard to keep four people with superpowers (one of them a giant rock-man) fighting a monster in the middle of the city under wraps, but if they weren't, it would seem weird to just all of a sudden have an established superhero team 40 something years before The Avengers without having any mention of them at all previously. I could see them working as a covert operations team with SHIELD much like Pym did.
I feel kind of the same way. Not that it doesn't conflicted with existing continuity, but does hurt the concept and theme about the world growing and being introduced to superheroes. If they are from the 60's I'd prefer they would get their powers while they were lost. So they return as super heroes.
Expanding on NotSoLongAgo's idea, I like the prolog to be about two married scientist, a brother who's a former air force pilot, and a best friend who is one of, if not, the first (depending on historical timing and accuracy) jewish astronauts in space. Embarking on the first private space venture to test the Richard's Near-Light Drive. A propulsion system that can approach the speed of light. The media dubs them "The Fantastic Four: The First-Family of Space Flight". A 10 year mission, fly for five years to the nearest star Alpha Centauri, they'll take multiple readings and send a signal back to earth before returning (relativity making time movie slower for them). They leave Earth in iconic blue space suit with the number four on it's chest and the world awaits their return. But decades come a go and world is left to wonder "What happened to the Fantastic Four?"
In transit (either on the way there or back) Their ship is bombarded with cosmic radiation, transforming them, damaging the ship, and causing the light drive to warp along the path of the radiation. Richards gains consciousness long enough to shut the drive off to then crash on a planet whose system is surrounded by the same cosmic radiation. The peaceful locals, who are advance but the presence of the radiation makes space travel and off planet communication difficult, helps them. Then we will time skip a year later.
That would be the first ten-fifteen minutes.
After year with the four live among them, Reed and Sue have adopted the language while Ben and Johnny just barely getting by. Reed along with local scientist develop clothing out of unstable molecule that can transforms with them. The Locals create four suits, all blue with a 4 on the chest (resembling the Ultimate F4) as a gift, symbolizing the spacesuits they were wearing when they first arrived. And still expanding on NotSoLongAgo's idea, one day another ship crash from encountering the same cosmic radiation present in the system. It is a Skrull scout ship with one passenger, Kl'rt. He initially disguises himself as a different race and convinces the locals and the four to help him fix his ship, promising to take them home. As the ship is finished and a shield is designed to protect against the cosmic radiation. It's revealed the Kl'rt is not only a Skrull, and the the local know the Skrull have been on a campaign of conquer habitable planets, but that he was also mutated by the radiation and due to his relative proximity to the four had mutated to have all their powers. The four manages to win due to Kl'rt's inexperience. The lock him up in the scout ship which they intend to use to find home or at least make their way to a space flight capable planet. They bid the peaceful locals a good bye and are off.
They find Skrull universal translators which luckily has english programed in (hint, hint), but unfortunately the ship is on pre-programed auto pilot to a Skrull warship. The four manage to stay hidden and learn more of the Skrull race and realized not only the danger the peaceful planet is in, but the danger should the skrulls discover the cosmic rays. The disable communication, battle the skrulls, including another epic fight with super skrull, they end up destroying the warship and escaping on a vessel to find the next world, but luckily Reed finds Earth in the navigation (another hint, hint). They begin their journey home, except the Super Skrull manged to get on the ship. They engage in a final fight that greatly damages the vessel, Super Skrull is defeated and thrown out into space to an apparent death. The ship crash lands outside of new york. By the time they come to, police and crowd have gathered around the ship, with the military and Avengers on they way. They are initially shocked by the Thing and Human Torch but once the see Reed and Sue the begin to realize, the Fantastic Four have finally come home... to Marvel.
The epilogue Reed uses the massive amounts of funding to build a new lab on top of his old home and work place, the Baxter building, and establishes the Fantastic Foundation. And begins work on studying the skrull ship.
Post credit scene, the Super Skrull survived by using a force field, he takes refuge in an abandoned building where he is surprised by a human who states "Worry not brother, for the invasion here, has already begun" The human transforms into a Skrull.
Not sure how to incorporate Doom as a F4 specific villain, maybe he was a student of Richards. Seeing getting lost in space as an idiotic failure dedicated himself to earth and comes to one conclusion. He must rule.
And I see the next big Avengers movie after this being Avenger: The Kree-Skrull War.