On April 18, NASA announced that the Kepler space telescope had discovered three new planets that might be capable of supporting life. Two are part of the stellar system Kepler-62 that now has five identified planets with masses ranging from 0.54 to 1.95 times the mass of Earth. The two newly discovered planets in the system are potentially capable of supporting life, as we know it.
Kepler-62 is 1,200 light-years from Earth. Even travelling at the speed of light it would take 1,200 years to get there as measured on Earth, so future colonization by earthlings when our planet becomes unlivable would seem unlikely.
However, it is not impossible. Imagine building a spaceship that can leave Earth and accelerate at a constant one g, the acceleration of gravity on Earth. The passengers and crew would be able to travel with all the comforts of home, without having to worry about their muscles atrophying.
Applying Einstein's theory of special relativity, after the spaceship has traveled a distance x from Earth, the elapsed time as measured by any clocks onboard the ship, including the body clocks of the human crew and passengers, is given by the equation s = acosh(gx + 1)/g, where acosh is the arc hyperbolic cosine.
Of course, the ship can't continue to accelerate, since it needs to be moving at the same speed as the planet when it arrives. When it reaches the halfway mark at 600 light-years from Earth, the crew turns the ship around so it then decelerates at one g for the remainder of the trip. Those onboard notice no change. They continue to experience the equivalent of gravity on Earth.
Plugging x = 600 light-years in the above formula, we find that 7 years have elapsed on the ship clocks at the halfway mark, although six centuries have passed on Earth. The ship travels for another 7 years and arrives at Kepler-62 with a crew and passengers that are 14 years older than they were when they left home while 1,200 years have gone by back on Earth.
Now, building a spaceship capable of accelerating at one g for 14 years is not within any known technology, but we can't prove it's impossible. However, while also not impossible, it is highly unlikely that the space travelers will find a planet orbiting Kepler-62 that they can simply beam down to like Captain Kirk and Mr. Spock, breath the air, find food, and generally settle down and build a colony.
News reports on the discovery of "habitable" planets are highly misleading. What scientists define as habitable just means that conditions may exist that are adequate for life based on carbon chemistry, such as the generic type of life we have on Earth. The chances that such life would be exactly like ours, with the same proteins and DNA, would seem very unlikely. Furthermore, finding a planet anywhere in the galaxy so closely resembling Earth that humans could live on it without life support strikes me as to close to impossible as anything that is possible can get.
In the meantime, our intrepid travelers at Kepler-62 might decide to turn around and go back to Earth. By that time a new generation would have been born onboard ship and they would arrive back on Earth 14 years later. There, 2,400 years would have elapsed. They would be in the same position as humans returning now after having left in the fifth century BCE, at the time of Socrates.
However, for the Keplerians, by 2400 CE Earth would probably have cleansed itself naturally of the detritus left behind as earthlings destroyed themselves. Perhaps the young Keplerians, born and raised in space will do a better job as they recolonize Earth and give humanity another chance.