Breen space was a region of space claimed as territory by the Breen Confederacy and later by the Breen Imperium.
According to Worf, the Breen "did not tolerate incursions into their space," while recalling the Klingon Empire's attempt to conquer their homeworld, only to never be heard from again. (DS9: "'Til Death Do Us Part")
In 2367, the Breen were know to have outposts in Sector 97. The black cluster located in the sector was considered to be "quite a distance" from both Romulan or Klingon territories. (TNG: "Hero Worship")
After Worf and Ezri Dax were picked up by the Breen on Goralis III in mid-2375, Dax wondered what they were doing at that planet, as she considered it to be "a long way" from Breen space. (DS9: "Penumbra") Three days later, while still aboard the Breen ship, Worf reckoned that they could be in Breen space by now. Dax, however, mused that it was also possible that they saw their shuttle craft and were taking them back to Deep Space 9. It was ultimately learned they were both wrong, and they were in fact en route to Cardassian space. (DS9: "'Til Death Do Us Part")
In the late 32nd century, Federation worlds near the border of Breen space expressed concern for their safety with the increased aggression of the Breen and the Breen Civil War. Federation Ambassador Saru embarked upon a diplomatic mission to several of these worlds in 3191 and upon his return, warned that the worlds were already on edge and the potential ramifications of Primarch Ruhn's death could cause panic. (DIS: "Jinaal", "Erigah", "Lagrange Point")