When Does the Film "Captain America: Civil War" (2016) Take Place?
"Captain America: Civil War" is a political superhero film written by Christopher Markus with Stephen McFeely [adapting "Civil War" #1-#7 (2006) by Mark Millar] that released April 27th, 2016. So when is "Civil War" set?
It takes place on the Marvel Cinematic Universe timeline in the year:
We know this because the Vision says, “In the 8 years since Mister Stark announced himself as Iron Man” and this certainly refers to the events of "Iron Man" (2008), which takes place in 2008 AD. The main action of the narrative is only labeled as "PRESENT DAY" by a relative time setting title card after the 1991 AD prologue's end but math since the events of "Iron Man" (2008) tells us when it is.

2016 AD
We know this because the Vision says, “In the 8 years since Mister Stark announced himself as Iron Man” and this certainly refers to the events of "Iron Man" (2008), which takes place in 2008 AD. The main action of the narrative is only labeled as "PRESENT DAY" by a relative time setting title card after the 1991 AD prologue's end but math since the events of "Iron Man" (2008) tells us when it is.
2008 + 8 = 2016.
More specifically, very early in the movie's main action, there is a "The Daily" newspaper front page on which the date cannot be read but an advertisement is visible informing its readers of the "SUNDAY MARCH 22" date of an event only called: "2016 ELECTION CONVERSATIONS". The story's context and the fact that this newspaper and the one next to it are both dated for a Saturday implies this newspaper is recent. Possibly even the very day the opening scene takes place. This means it's very likely to be sometime in late February or the middle of March of 2016 AD.
A TV news reporter refers to this early part of the film as "last month" so everything after the first eighth of the movie must be in March or April of 2016 AD.
Additionally, there is a prologue clearly labeled by a time setting title card reading "1991". This is confirmed as this prologue takes place just before and after the 1991 AD flashback near the end of the plot.
There's also a flashback revealing more about the setting during and just after the 1991 AD prologue.
Finally, there's another flashback near the end of the narrative that must take place in December of 1991 AD as the VHS tape that frames it is clearly dated for "DEC. 16 1991". (In fact, a newspaper front page dated for "FRIDAY, DECEMBER 17, 1991" describing this exact event is visible on screen during a slideshow presentation in "Iron Man" (2008) for a quick moment.)
This Geekritique article and video examines its exact setting.
It is a chronologically midway film in the Marvel Cinematic Universe.