The Past, the Future, and Beyond
With the exception of the first scene of the hominids, the setting for the year 2001: A Space Odyssey is the year 2001. Except not really.
Our universe's 2001 was a very different affair. The year didn't bring us commercial space flights, research laboratories on the moon, or space stations with artificial gravity that allowed you to enjoy a drink in the Earthlight room. We were stuck with unglamorous space shuttles and the International Space Station.
It did, however, provide some very important moments in world history: Wikipedia went online, the World Trade Centers were attacked, Enron filed for Chapter 11 and the iPod was released. Then there was Michael Bay's Pearl Harbor and the Backstreet Boys' only performance with N*SYNC… okay, so 2001 had some not-so-important moments, too.
The Final Frontier of Yesteryear
Here's the way to think of the year 2001 in this film. It isn't the future; it's the future as seen from the past. As such, we need to consider what aspects of the 60s paved the way to the vision of 2001 as seen by Clarke and Kubrick.
In October 1957 the Russians launched the world's first satellite, Sputnik, into space, and America, currently embroiled in the Cold War with the USSR, did not take the surprise well. After all, if Russia could launch a satellite into space, then what else? Nuclear warheads? Postage? This clearly could not stand, so in 1958 the U.S. launched its own satellite, Explorer I, and the space race was on like Donkey Kong, erm, Space Invaders.
In true "my stuff is better than yours" fashion, America and the Soviet Union were competing to see who could out-outer space the other, and the Russians were handily whooping on the U.S. The Soviets were the first to smack the moon in the face with a space probe, Luna 2, and also the first country to have a man orbit the Earth, one Yuri Gagarin.
These events led President Kennedy to state in 1961 that the U.S. would put a man on the moon by the end of the decade. NASA's budget received a welcome 500 percent bump and the organization's staff boomed. The result? In 1969, the Apollo 11 mission landed on the moon, effectively "winning" the space race for the U.S. (Source) Some of us have been regretting it ever since.
This is the climate that produced 2001. It was an optimistic time. Based on the advancements in space technology made during the 60s, Kubrick and his team thought it feasible that commercial flights would take visitors to the moon and back (or, for that matter, that Pan-Am would still be a thing). Space stations would serve as way stations for travelers, and people would work, if not live, on the moon by 2001. They predicted that people would play games with computers. Poole playing chess with HAL is something we don't give a second thought to today, but back then it was a revolutionary idea—not even Atari had come to market yet.
Science fiction trappings that weren't feasible, or downright impossible—such as travel faster than the speed of light and ships that produce gravity without centrifugal force—were left out of Kubrick's film. And while we obviously didn't hit all the milestones set by the film, it still presents a possible future for us living in the years well beyond 2001.
Evolution or Bust
Kubrick and Co.'s vision of the year 2001 is only part of the setting. We still have to incorporate Africa into the equation.
The film opens in Africa right at the Dawn of Man. Humanity as we know it isn't around yet, but our early hominid ancestors are still kicking about the savanna. Their day-to-day existence includes eating, hanging out, and trying not to die. That's how it is until a monolith comes, tweaks them in some unknown way, and they eventually develop the ability to experiment and create tools. Most anthropologists believe that Africa was, in fact, the cradle of mankind, minus the part about the monolith.
The monolith jump-started the evolution of the hominids into humans, and so the setting of Africa sets the stage for major themes in the film: evolution and scientific exploration. Ultimately, the space setting also connects with these themes, making the history of human evolution part of the setting.
By starting in Africa and moving us into space, the film follows humanity's evolution from beginning to end. Sure, they skip a few chapters of the story, a couple million years' worth, but it's in space that humanity evolves to its next stage, the Star Child. To the Star Child, space is home, a new setting for all of humanity's future odysseys.