Divergence

One of the most common misconceptions about the Fallout world is that it is just the future of our own world and that it works just like ours. Another misinterpretation is that the Fallout world is the result of a nuclear war that occurred in the 1950s of our timeline. Both of these ideas are entirely false.

The Fallout world is historically divergent from our own and also is fundamentally different in terms of how science works. The base concept for the setting is 1950s World of Tomorrow after the bomb. This means that before the war, the Fallout world was more or less what the people of the 1950s thought things would be like in 2077. This means housecleaning robots and laser guns were the norm. The only difference with conventional World of Tomorrow stuff was the absence of any space exploration. In the Fallout world, the Space Race of the 1960s never occurred. Consequently, miniaturization of computers was never necessary and the transistor was never invented. Instead, humanity invented fusion power. This is why the computers in Fallout are all of the old reel to reel and vacuum tube type and are generally very large and bulky. Even if these computers are very advanced, the technology to make them smaller never existed. Don't ask about the PipBoy, though. They're probably just really small vacuum tubes. The exact historical details of the divergence and even the exact moment when it occurred are unknown and irrelevant. All that's important is that this alternate universe eventually spawned the current Fallout world.

Fusion power, however, allowed the Fallout world to do really neat things right here on Earth. A clean, renewable, plentiful and portable source of power meant that things like power armor and energy weapons could be built, as well as all the housekeeping robots.

Another relevant fact is that the lack of a Space Race means, of course, no rockets. This also means no missiles. In Fallout, there are no big red buttons or ICBMs. All the nuclear weapons delivered in the Great War were bombs dropped by planes (as an aside, jets do exist in Fallout since the jet engine was invented shortly before World War II). No one knows what happened to the pilots of these planes after they dropped their payload. It's probable that most of them just disappeared as their airbases were likely high priority targets for enemy aircraft.

Arms & Equipment
Another very important thing about the divergence of Fallout's history from our own is that it means that pretty much nothing as we know it exists in Fallout. This is why real world weapons in Fallout games don't work. How can, say, the M24 Sniper Rifle exist in Fallout when it was first introduced in a 1960 that never happened in the Fallout universe? It can't, really. Thus, Fallout substituted fictional weapons. For instance, instead of including a 9mm Beretta as a basic handgun, Fallout gave players a generic 10mm pistol. Instead of the aforementioned M24, Fallout had a generic sniper rifle. Looking at the detailed descriptions gave names to the weapons (the sniper rifle, for instance, was the DKS-501), which were entirely fictional. The single exception to this was the .44 Desert Eagle handgun, which exists in our timeline and shouldn't really have appeared in Fallout. The many real world weapons that appeared in Fallout 2 were also errors. The advantage with using fictional weapons, beyond simply respecting the timeline divergence, is that it allows designers to create the weapon they need to fit the game. Inserting real world weapons only leads to firearm experts complaining about inaccuracies in how they're implemented and prickly fans complaining about the presence of these impossible weapons.

Given that the historical divergence occurred soon after World War II, it is possible that war era weapons would exist in Fallout. That said, given that a weapon manufactured in 1941 would be 220 years old by the time Fallout began, finding a functioning wartime weapon is unlikely.

Physics in a Different World
The Fallout world doesn't only diverge historically. The universe itself is fundamentally different. The World of Tomorrow theme doesn't only apply to what technologies exist and how history unfolded. It also applies to the laws of physics. In our world, we know that exposure to radiation just kills you. In the Fallout world, however, severe radiation isn't always fatal, and it occasionally produces unlikely or impossible mutations including increased size and in the case of ghouls, extremely long life coupled with a decaying body. Remember classic movies like Them or The Fifty Foot Woman, where freak nuclear accidents caused giant ants or people to appear? That's how it works in Fallout. All science behaves the way it was popularly thought to behave in the 1950s. Similarly, this means that there was no nuclear winter in the Fallout universe since that theory was put forward in our 1970s and only concretized by Carl Sagan and a team of researchers in our 1983. Whenever considering science in Fallout, remember this rule: Science!, not science.