Answer by Robert Frost, engineer and instructor at NASA:
Let's ignore the technologies that might be impossible (e.g., warp drive, dilithium crystals, and transporters). Let's ignore the technologies that we have no idea how to reproduce in a similar way (artificial gravity). Let's just focus on trying to build a space-worthy scale replica of the USS Enterprise that uses existing structural and propulsion capabilities.
The Starship Enterprise is much more massive than the International Space Station. Here's a comparison of their respective sizes. (Note that I've used the original Enterprise, not the Abrams version, which is considerably larger.)