Romulan and Klingon space, which happens to be on the other side of the Federation from Cardassia
Does it? Klingons once had a border quarrel with Cardassia, at the Betreka Nebula ("Way of the Warrior").
And while there are no direct references to Romulans sharing a border with Cardassia (just to Romulans operating at the "Cardassian border", which might be their border against neutral space, in "Improbable Cause"), there are indications of ongoing Cardassian/Romulan intrigue (say, Terok Nor featuring Romulan components in "Dax" - technological aid or espionage?). And "Birthright" has Worf take a ride in a small and thus supposedly slowish craft from DS9 all the way to Romulan borderlands, suggesting spatial proximity. So the three villain realms might actually be within easy travel of each other, with only sparsely populated parts of the UFP in between at most, and we'd be none the wiser about the ability of the UFP sensor systems to monitor intrusions into "UFP proper".
has a very thorough analysis on the size of the Federation, and concludes that, while Cardassia and Romulan/Klingon space are on opposite sides of the Federation, they aren't really that far away. Also, the Romulans and Cardassians are about as close as they can get without being right next to each other.
It is also important to note that space is big. It is probably really easy to cross Federation space, no matter who you are, because even if you can be easily seen, someone has to be looking at you for something to happen. Starfleet has a lot of observation resources at its borders with its more historically hostile neighbors, but not likely too much pointed inward.
For what it's worth, this also more-or-less matches to the map in Star Trek Online, which seems to be a lot more under-the-control of CBS than your other typical non-canon sources.
There is a canon map (the one in O'Brian's classroom on DS9) that shows that indeed Cardassia and Romulus (and Qo'nos) are indeed on opposite sides of the Federation; however, they are far too far apart to be quickly reachable without some other stuff we don't canonically know about (like subspace "lanes" where your warp relative to outside observers might be faster than what your ship can do - which would explain a lot