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Multiple realities
(covers information from several alternate timelines)

The Federation-Klingon border was the division between the United Federation of Planets and the Klingon Empire.

As of 2268, Deep Space K-7 was located near the border, only 3.26 light years from the nearest Klingon outpost. According to Pavel Chekov, this was close enough to even "smell them [the Klingons]". (TOS: "The Trouble with Tribbles"; DS9: "Trials and Tribble-ations")

In the late 23rd century, a Neutral Zone existed along the Federation-Klingon border. In preparation of a new peace conference in 2293, the dismantling of the Federation's space stations and starbases along its course was discussed. (Star Trek II: The Wrath of Khan, Star Trek III: The Search for Spock, Star Trek IV: The Voyage Home, Star Trek VI: The Undiscovered Country)

The abolition of the Klingon neutral zone was likely part of the Khitomer Accords of 2293. Its establishment, on the other hand, might have been part of the Organian Peace Treaty of 2267.

The distance from Earth to the border was never clearly stated, though Kirk mentioned in Star Trek VI: The Undiscovered Country, while the USS Enterprise-A was escorting Kronos One from the Klingon border to Earth, that being a thousand light years away from Federation Headquarters allowed them to enjoy Romulan ale even though it was illegal.

In 2369, Governor Torak was in charge of a region of space near the Federation-Klingon border. One of his ships, the IKS Qu'Vat under the command of Commander Morag, patrolled the border, passing near the Federation Relay Station 47 every six days. (TNG: "Aquiel")

In late 2372, following Chancellor Gowron's demand for the Federation to leave the Archanis sector, several representatives from planets on the Federation side of the border urged a preemptive strike on the Klingons. (DS9: "Broken Link")

The Narada emerged in the alternate reality in 2233 next to a star 75,000 kilometers from the border. It attacked and destroyed the USS Kelvin, which was sent to investigate the anomaly. (Star Trek)

The significance of the location is revealed in deleted scenes where Klingon warbirds capture the Narada after it collided with the Kelvin, accusing Nero of trespassing.