Star Trek takes place in our galaxy, in the late 24th century. (with the most-distant movie set in 2379.) The Milky Way Galaxy is absurdly big -- some 120,000 light-years in diameter, with a center some 27,000 light years away from Earth. The nearest galaxy is the Small Magellanic Cloud, some 200,000 light-years away.
Since Voyager found themselves 70,000 light-years from home, and projected their time to get home at 75 years, we can infer an effective cruising speed of 933 ly/y, or 2-3 ly/day. Considering that they weren't stocked for it, and had to scavenge along the way, one can presume that a well-stocked round-trip vessel with custom-designed engines would, at best, match their speed.
Taken together, that means that a Federation Expedition to reach the closest galaxy and return home would take over 400 years. Which means that, if they left today with 24th-century technology, such an expedition wouldn't return until some thirty years after the star trek canon ends.
So, Star Trek's "universe" is really just a portion of our current galaxy.
It's worth noting that, although not closely followed in the show itself, the official star trek separation of "quadrants" uses a meridian pretty close to Sol, the star around which Earth orbits. So, the federation encompasses some territory in both the "alpha" and "beta" quadrants, as do the major and minor powers with which it can negotiate, plus some special-case exploration to the "delta" and "gamma" quadrants, from Voyager's journey plus the DS9 wormhole.
Of course, this only explores the Federation's domain of exploration and influence. Several other powers, such as the Borg, the Fluidic-space aliens, and cosmic-level powers such as Q have unmeasured spheres of dominion, and have probable access to whole galaxies that the Federation does not.
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