When is Legion happening?

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.

However, the fact that Amy (David's sister) want to contact her husband by email (see these subs at 39:24) makes me wonder if the world is in another timeline or if there is an error in the script.

In terms of the X-Men cinematic universe, Legion does take place in separate parallel universe. So like the previous answer states, it also fixates on David's persepctive of reality allowing creative freedom and not always being able to pin down "when" this takes place.

However, his reality or expiriences are not the only ones on display. We see flashbacks (that look very 30s-50's) from Ptonomy for instance, which are his memories. Also in the last few episodes, viewers get to see the other characters exprience David's mind--so there is an "outside looking in" aspect as well (it's not ALL in David's mind). Then there is also the astral plane---all of these things make Legion metaphysically confusing to discuss, but the point is the concept of identity or reality is not limited to any one thing.

Landgraf stated, in January 2016, that the series would be set in a universe parallel to the X-Men films where "the US government is in the early days of being aware that something called mutants exist but the public is not".

The producers hoped that audiences would watch the show because of its character-focus and the talents of Hawley and the cast, rather than to "see a Marvel franchise show."[57]Donner also noted that having Hawley focus on Haller's perspective of reality rather than connections to the X-Men films allowed the series to avoid the convoluted continuity of the films, "because we play with so many different timelines, and we rebooted and not really rebooted and all that" throughout the films. Therefore, "the cinematic universe will not worry about Legion. They will not worry about these TV worlds at all. https://en.wikipedia.org/wiki/Legion_(TV_series)

“It’s not in the continuity of those films in the sense that the current X-Men films take place in a universe where everybody on planet earth is aware of the existence of mutants. Legion takes place in a parallel universe if you will where the government is aware mutants exist but the public is not. I wouldn’t see characters moving back and forth because they really are parallel universes.”http://www.slashfilm.com/x-men-tv-series-legion/