Corrected entry: The whole plot centers on the crowded conditions of the planet. The people are literally bumping into each other. Even if they had the blueprints to build a full size model of the Enterprise, there would be no room for it.
Corrected entry: How the heck did the Gideons (who are not members of the Federation) get such exact specifications to make what Spock describes as "an exact duplicate of the Enterprise"? Does Starfleet just hand the plans out to anyone?
Correction: This is an unanswered question, not a plot hole. They could very well have advanced scanning capabilities. In fact, that would be a more realistic explanation than schematics, as personal items are duplicated as well.
Correction: That's actually a plot point of the episode; that they were so desperate for this to work they were willing to sacrifice that much space.