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: I've used the original Enterprise, not the Abrams version, which is considerably larger).
The International Space Station (ISS) is the largest, most complex international scientific project in history. For decades, studies were done that developed the ideas and technologies needed to build it (e.g. the steering law algorithm was written during Apollo). The prime contractor (Boeing) was selected in 1993. The first module launched in 1998. Over the following decade, 15 countries invested their treasure (~$135 billion) to build the ISS. It took 26 Space Shuttle flights to deliver the US segment modules. An additional Russian rocket launch was needed for each Russian module. Over 1000 hours of EVA were needed for the assembly.
It would take hundreds of launches just to get the materials needed for the Enterprise into space.
I doubt it. To read more, click here.