Two thousand years ago, a writer envisioned a trip to the Moon. In his time, people thought such a trip impossible and it would be improbable that humans would ever have the technology to do the trip. I think it's premature to say whether or not we can travel to the nearest stars. It may be that in another two thousand years that humans may have traveled to the nearest stars.
I think that space travel to the nearest stars may be a series of leaps, like what the Polynesians did when exploring the Pacific. They would travel to one island, settle that island, and travel to another island. I see humans doing that in space.
One of the reasons I am hopeful is that humans do have a tendency to be slow to move into other areas. It took tens of thousands of years for our species to move out of Africa into the Sinai peninsula, and eventually to the rest of the world.
I am doubtful that humans may colonize as much space as was shown in Star Trek or Doctor Who, where humans have spread out into systems that are thousands of light years of Earth. I think that it is more likely that humans will stay pretty close to the home system.
The benefit of the discoveries we are now witnessing in astronomy is that we are building a list of probable systems that humans could travel to and colonize. And, now with the discovery of a planet in the Alpha Centauri system, I think it's possible to say that there might be other planets in the system. So, when we do get there, humans may find a world to colonize. So, I think it possible that in one way, Star Trek might be prophetic - in "Return to Tomorrow", Captain Kirk said that humans went to the moon, went to Mars, and went to Alpha Centauri. I can see that progression happening within the next two thousand years.