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Although there have been numerous prop maps developed for use on Star Trek sets, there has never been an intentionally definitive map developed for either the writers or for use as a prop for understandable reasons. As such, the boundaries and even the sizes and arrangements of the major political entities in Star Trek have never been consistent. Oddly enough, however, the operating ranges of Starfleet starships seems to shrink with each incarnation of Trek until Enterprise - The Original Series regarded the entire Milky Way as within the reach of Starfleet, while Deep Space Nine restricted Starfleet to the Alpha Quadrant - and Voyager reduced intraquadrant travel to a crawl (although this seems to apply only to Alpha Quadrant starships in the Delta Quadrant).
Because I first became a Trek fan a few years before "The Next Generation" first aired, I tend to place a lot of importance on the TOS-era estimates of the Federation�s size. The map below is my own take on the political boundaries of the Milky Way Galaxy, circa 2375. It was inspired by the map written up by Franz Joseph for his "Star Fleet Technical Manual" and by Graham Kennedy's own attempts at Star Trek maps.
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In creating this map (whose scale is in thousands of light years), I followed a number of assumptions:
There are multiple "levels" of "Federation Space." The 150+ full members of the Federation (Earth, Vulcan, Betazed, etc.) along with Starfleet's Spacedocks are within an area 8,000 light years across that is well removed from the other Alpha Quadrant powers (and is very well protected). A much larger area (based on Franz Joseph's published TOS-era map) some 31,000 LY across contains the Federation's 700+ Starbases and its major colonies, and is routinely patrolled and charted by starships. The outermost (and most recently claimed) volume of space by the Federation is its frontier - which has a number of small outlying colonies and the Federation's outposts and "Deep Space" space stations. Overall, Federation space is a somewhat comma-shaped region some 35,500 light years across in the coreward axis and up to 41,000 light years across at its widest point (with the Romulan Star Empire inside of the 'crook' and the Cardassians and Ferengi on the other side the 'tail') that is 12.5% of the Milky Way Galaxy - and the practicality of that assessment is addressed here.
The Federation, as well as the Klingons and Romulans, are entirely within in the Alpha Quadrant, and the latter two are in close proximity with every other major power in the DS9 series. Not only does this give more of a "Alpha vs. Gamma Quadrant" sense to the Dominion War, but it makes feasible the ease with which the Klingons and Romulans can reach DS9 (while not conflicting with the large size estimate of the Federation) - and explains why the Federation is the only Alpha Quadrant power to have been openly attacked by the Borg.
Travel within a Quadrant is usually treated by TNG and DS9 episode writers as a far shorter prospect than travel between Quadrants - my guess is that the Quadrants are roughly defined - and separated - by 670 light-year stretches of space where warp drive operates far slower than within a Quadrant (go here for more information). Nonetheless, during the late 23rd Century, Starfleet was able to locate a number of spaceways that linked the Alpha and Beta Quadrants, thus enabling Starfleet to begin exploring the Beta Quadrant as early as the 2280s. By 2364, a new generation of high-speed exploratory vessels, combined with improved techniques and equipment for detecting and charting spaceways and an extensive charting of the spaceways crossing the Alpha-Beta border, enabled Starfleet to chart the Alpha and Beta Quadrants at the combined rate of over 6% of the Milky Way Galaxy per year ("Where No One Has Gone Before," "The Dauphin" [TNG]), resulting in the nearly complete charting of both quadrants by the time the Bajoran wormhole was discovered in 2369.
The shape and extent of the Dominion is based on Graham Kennedy's estimate, but I have moved the Dominion's borders some distance away from the wormhole exit, to reflect how the wormhole was used for two years before the first encounters with the Dominion. As for Federation-claimed space being much larger than the Dominion, in "Sacrifice of Angels" [DS9], Weyoun openly fretted to Dukat about the anticipated difficulties of "holding onto a prize as vast as the Federation" - which, I hardly think would have been a concern unless the Federation was so much larger than the Dominion as to pose unprecedented difficulties in pacification (in which the Vorta and the Jem'Hadar have a lot of experience in).
The shape and extent of Borg Space is likewise based on Mr. Kennedy's own estimate, as it allows for the Borg encounters that Voyager's writers are so fond of. Nonetheless, I've elected to not only shrink the extent of Borg space (primarily to keep the Borg a Delta Quadrant race), but to split Borg space into two parts in order to allow for how Species 8472, which was invading the Collective in regions close to the route Voyager was taking through Borg space, could be on the verge of obliterating the Borg while appearing to remain at the fringes of Borg space. The part deepest in the Delta Quadrant is what I regard as the Collective proper (which contains the Collective's unicomplexes and no unassimilated inhabited planets) - I guessed the subpart would resemble the Borg Collective's insignia (with Unimatrix 01 placed at the center "palm" of the symbol). The much larger section within the Delta Quadrant is what is covered by the Collective's five transwarp hubs outside of the Collective proper - here, Borg ships occasionally raid the unfortunate inhabitants, but have not yet bothered to establish a permanent presence. As for the Borg concentrating on coreward races of the Delta and leaving the rimward races mostly alone - given the rimward Delta Quadrant races we've seen on Voyager (ie. Kazons, Vidiians), can you blame them? <g>
Note: the galaxy-spanning transwarp hub used by Voyager in "Endgame" [VOY] had to have been an atypically large, recently constructed example that was not yet fully operational at the time of its destruction in 2377 - else the possession of a permanent transwarp aperture right at Earth's doorstep would have obviated the need for the Borg to fight their way through the rest of the Federation in "The Best of Both Worlds" [TNG] and "First Contact."