At most, Doctor Strange is set over a period of 3 and a half years.
Between out-of-universe information and in-universe references, we can safely assume that between the start of the movie and the first end credits scene, no more than 3 and a half years have elapsed.
The first thing we need to do is ignore a reference to an experimental armour that we hear before Stephen Strange crashes his car. Director Scott Derrickson recently confirmed that this is not meant to be Colonel James "Rhodey" Rhodes, meaning that we cannot use this as evidence for when the film is set.
This answer from @phantom42 includes a link to an interview with Kevin Feige that confirms that at the time of Captain America: The Winter Soldier, Stephen Strange is just a highly talented neurosurgeon.
"He is very well-known as a surgeon," Feige said during a recent visit to the set of Doctor Strange. "He has got various awards and plaques you're going to see over there; he attends various galas, might be driving at one point in this movie. He has name recognition and a talent and certain Hydra computers identified him as somebody that could cause trouble for their agendas."
So the answer is, Strange wasn't an immediate threat to Hydra in any meaningful way, but as a brilliant and famous person whose agenda didn't mesh with theirs, he stood a high probability of being a problem for them if they successfully took over government. Think of it like a politician targeting Bruce Springsteen or Ted Nugent because they might use their concerts as a platform to spread a message the politician in question didn't like.
This conclusively proves that as of spring 2014, Stephen Strange has not yet had his car crash. This means that we know for sure that he does not start his training until some point after spring 2014.
Our "end point" comes in the form of a post-credits scene in which;
Doctor Strange offers to help Thor and Loki find Odin on Earth.
This is a teaser for;
Thor Ragnarok, which is set to come out in autumn 2017.
Assuming that the film above is set in roughly the same time period that it is released (as tends to be the case with most films set in the Marvel Cinematic Universe), then we know for sure that Doctor Strange ends by autumn 2017.
As such, at the very most, Doctor Strange takes place between spring 2014 and autumn 2017, a period of 3 and a half years. It's worth noting that there are some people reporting that they saw trophies/awards in his apartment at the start of the movie dated 2016 (which would obviously severely reduce the period the film is set over to around about 18 months), but as I cannot verify those claims I have not taken them into account in this answer.
As for the word of God answer, when asked about this on Twitter, director Scott Derrickson simply replied;
"Time is relative"
