One of the benefits of an animated series is the creators have greater freedom to add unexpected -- and oftentimes impossible -- elements to storylines. This extends very much to Archer, which at times feels like the ultimate hodgepodge of cultural references and spy technology. Considering all the futuristic tech, retro styling and contemporary culture in the FXX series, it is difficult to figure out just what year Archer takes place in.
Archer was designed with a certain aesthetic in mind as a parody of the great spy films of the 1960s, including the early James Bond movies. The general aesthetics of the ISIS offices were designed with a beige color scheme and technology that wouldn't be out of place in that period. The computers are larger devices, with retro operating systems. However, Archer also operates with a sliding time-scale in terms of character history and available technology. There is a mix of cars and other vehicles that often range from the 1950s to the 1980s.
At the same time, there are also a number of modern inventions and technologies, such as the characters all having easy access to the internet and cellphones. Some of the futuristic technologies available even suggest the show could happen sometime in the near future, as exemplified by the existence of a fully operational Sealab station in Season 4's "Sea Tunt" or the impressive space station featured in Season 3's "Space Race." Notably, the Soviet Union is still a major global power, which fell apart in 1991 in the real world. However, the Soviet Union is often shown to be waning in terms of potential and its citizens are not shown enjoying easy lives, implying the series takes place closer to the collapse of the alliance.
The given ages of the characters make a concrete timeline even harder to figure out. The first episode of the series, "Mole Hunt," specifically states that Archer is 35 years-old. However, his mother's activities in the OSS and working in a post-WWII European theatre suggest the show would actually take place in the 1970s. Other elements of Archer's backstory hint he should be even older. Specifically, his valet Woodhouse was revealed to have been born in 1894, allowing him to serve in World War I and explaining his age. However, Season 4's "Once Bitten" had a near-dead Archer face many of his earlier life choices, including a 1920s depiction of New York City, where a teenaged Archer was able to prove his athletic abilities by playing against baseball stars of the time.
The biggest hiccup in terms of placing the show in any of these time-periods is that Archer is full of references to music, film and literature from across the 20th and 21st century. Archer often references elements of cultural touchstones such as X-Men comics and films like Smokey and the Bandit, which Archer apparently considered a favorite film in his youth, placing the show more or less in the present day. Perhaps most importantly, Archer has a keen awareness of the modern-day politics of gender, race and sexuality, with the cast representing a diverse collection of people, including the African-American Lana, the openly gay Ray and the sexually fluid Pam).
While the show is open about having a fluid timeline -- and has even poked fun at that element in some episodes -- it also gives Archer a wider breadth of potential storylines to play with. So long as the timeline of the specific characters remains consistent, then having no set year actually works in the favor of the spy comedy.