When Does the TV Show "Constantine" (2014) Take Place?
"Constantine" is a horror television series showran by Daniel Cerone with David S. Goyer [adapting "Hellblazer" #1-#2 v1 (1987) and "Hellblazer" #4 v1 (1987), all by Jamie Delano, as well as "Hellblazer: All His Engines" (2005) by Mike Carey] that first released October 24th, 2014. So when is "Constantine" set?
It takes place on an alternate timeline of the fictional DC Films timeline in the year:
We know this because in episode 2 "The Darkness Beneath", John's "UK" "DRIVING LICENCE" says it's valid from "01-12-06" to "01-12-16". So it's probably after January 12th, 2006 AD and before January 12th, 2016 AD, though Constantine doesn't quite seem like the kind of bloke to worry too much about being current with the letter of the law. In episode 5 "Danse Vaudou" Jon Constantine and Papa Midnite need to exhume a dead body and inside the mausoleum the engraved memorial plaque clearly reads "CLARKE DEVEREAUX 1976-2011". So it must be during or after 2011 AD but probably before 2016 AD.

2014 AD
We know this because in episode 2 "The Darkness Beneath", John's "UK" "DRIVING LICENCE" says it's valid from "01-12-06" to "01-12-16". So it's probably after January 12th, 2006 AD and before January 12th, 2016 AD, though Constantine doesn't quite seem like the kind of bloke to worry too much about being current with the letter of the law. In episode 5 "Danse Vaudou" Jon Constantine and Papa Midnite need to exhume a dead body and inside the mausoleum the engraved memorial plaque clearly reads "CLARKE DEVEREAUX 1976-2011". So it must be during or after 2011 AD but probably before 2016 AD.
More specifically, near the end of episode 3 "The Devil's Vinyl", during the showdown at the Scott University WIXA FM college radio station, there's a large poster for real-world band "WE ARE WIRES EP RELEASE PARTY" advertised as happening "Saturday, MARCH 22 {2014}" and, while it's possible this poster has been left up by lazy college students after the event has happened, it's probably just before. So the show starts in early 2014 AD.
In episode 7 "Blessed are the Damned" the Jefferson County (Kentucky) Sheriff's dashboard camera records the current moment and the timestamp in the bottom left corner reads "12/5/14" and bottom right "2:32 PM" for 2:30 in the afternoon December 5th, 2014 AD. So halfway through the season it's December of 2014 AD.
Additionally, episode 3 "The Devil's Vinyl" also includes a short three-part flashback to the Blues musician Willie Cole [based on the real-world Blues musician Robert Johnson (1911-1938)] recording a session of music. During the main sequence the blank vinyl record being made can be seen and it is clearly dated "6-18-38" for June 18th, 1938 AD. The real-world Johnson died on August 16th, 1938 AD so it's almost certainly June of 1938 AD.
Lastly, episode 10 "Quid Pro Quo" includes another three-part flashback spread out over the plot. The first part of opens with a time setting title card reading "Two Years Earlier". This would make the flashbacks take place in 2012 AD, though there appears to be no way to confirm this via context. A front page of the fictional "N.Y. JOURNAL-CONSTITUTION" in the third part tells us the events of the first part was "last weekend" so it's been a few days to a week and, as long it wasn't December 30th, the three parts are in the same calendar year.
2014 - 2 = 2012.
A fourth and a fifth flashback brings the viewer closer to the present at the beginning of the series so it's sometime during or after 2012 AD and sometime during or before 2014 AD during both. They are otherwise unplaceable.
It is an early chronological series in the DC Films multiverse timeline (a first chronological show to feature John Constantine) and can be watched either at its timeline point or before "Legends of Tomorrow" season three (2017) as an extended flashback.