In theory, the show is set in "modern day", similar to the X-Men movies and most other superhero TV shows. There's enough tidbits of modern life to put a lower bound on the time period, including the references to email you mentioned. (This might become relevant if, at any point in the future, the show is able to cross over with the movies, which is "not being ruled out" for now.)
However, the director has purposefully set up and shot the episodes in a way that makes it incredibly hard to pick out a time period. The ambiguity is part of the central theme of the show, which is that almost nothing we see on the show is real life. For the most part, what we see is David's impression of real life, and David is an unreliable narrator at best. That's doesn't just include the scenes where he's obviously hallucinating, but even the mundane parts of the show are being filtered through David's mind and his memories.
As Noah Hawley explained to EW:
When I wrote the script I assumed it was set in present day and in our world, and I think the network assumed that too. Then when it came time to make it I thought about it more as a fable on some level and I realized I wanted to make something subjective. Which is to say this whole show is not the world, it’s David’s experience of the world. He’s piecing his world together from nostalgia and memory and the world becomes that.
Anything that would serve to date the show is omitted as much as possible. The decor is old 70's and 80's decor but in places where you might believably see that decor still around. There are very few cars in the show, since the models would give things away. People rarely talk on the phone, except again where you'd expect to see really old phones still hanging around.
On a side note: the overall look of the show was heavily influenced by the visuals of A Clockwork Orange, which is set in the future but looks like it's set in the 60s or 70s.