The Gamma Hydra sector was a region of space. The sector contained the Gamma Hydra system. The Romulan Neutral Zone and the Klingon-Federation border were, also, in this sector. Sections 14 and 15 of the sector were in Federation space, while section 10 was within the Federation-Klingon border. (TOS: "The Deadly Years"; Star Trek II: The Wrath of Khan)
In 2285, the Starfleet Academy Kobayashi Maru exam was a simulated no-win scenario where the USS Enterprise was on a trainng mission in this sector and receives a distress call from the Kobayashi Maru. (Star Trek II: The Wrath of Khan)
Holodeck Simulation - Captain Riker Edit
In 2367, while exploring a cavern on Alpha Onias III, Commander William T. Riker was rendered unconscious by gases. While unconscious, neural scanners scanned Riker's brain. The scanners used elements of Riker's reality and constructed a holodeck simulation with those elements interspersed throughout, so that it felt real to him. What Riker wanted, the scanners made possible.
In one of these simulations, the Romulans launched the Praetor Campaign, which was a preemptive strike into the Gamma Hydra sector. The USS Titan, under the command of Captain Riker, detected and prevented this strike. At the conclusion of this action, on stardate 51692, Riker was awarded the Starfleet Medal of Honor and the Starfleet Citation for Conspicuous Gallantry. In 2383, this sector was mentioned in Riker's service record that was on display in his quarters on the Enterprise-D. (TNG: "Future Imperfect", okudagram)