An Intrepid or a Defiant would be easier to do. A Defiant especially so in fact.
I think the easiest way to build them would be to have them built in an actual shipyard. The type that builds marine vessels. They're already set up to throw together huge steel structures that are markedly different from ordinary buildings in form.
Perhaps the ST mockup could then actually be built mounted on stanchions above a stripped-down supertanker hull to carry it around, so it could actually travel about the world and dock in cities here and there.
Another idea... for a ST ship that could actually fly about, would be a giant rigid airship in the shape of the ship. With an aluminium skeleton holding the shape and the outer skin painted with all the markings in their appropriate places. It could actually be controlled from a replica of the ship's actual bridge in the series.
While this would create a pretty realistic looking "starship" that could fly around, only a very few interior parts could be recreated, since most of its internal volume would have to be given over to helium.
On an entirely different scale, there's this guy's build project:
R/C USS Reliant
I've seen R/C enterprises as well. The saucer shape of UFP vessels is actually pretty useful as an airfoil for making a flying R/C model.