Except that ships in ST always travel at the speed of plot. Travel time has never been a remotely reliable indicator of distance. Hell, three movies later it took them less than half an hour to reach the center of the galaxy. Even if we ignore that (yes, please, let's ignore that), in the sixth movie it doesn't take long at all for the Enterprise to get from Earth to the Klingon border and for the arrested Kirk and McCoy to subsequently be taken to Kronos (one assumes) for trial and then shipped to Rura Penthe. So by those standards, a 3-day journey could represent a pretty significant distance.
Yes, let's ignore the trip to the center of the galaxy in TFF.
In any event, if we're to believe the various maps floating around the internet, a number of them depict Earth as being near the border between the Alpha and Beta Quadrants, with Klingon space located only a short distance into the Beta Quadrant. Depending on whether the map is based on twenty third or twenty fourth century boundaries, the Klingon Neutral Zone separates Klingon space from Federation space. Romulan space is just "north" of Klingon space and is also separated from the Federation by the Neutral Zone.
I'm inclined to believe the arrangements of the three powers relative to one another given the events of TUC and The Next Generation
. Camp Khitomer was located near the Romulan border, and the numerous skirmishes between the Klingons and Romulans in TNG suggest that the two species were Beta Quadrant neighbors.