Is interstellar flight possible in the near future in a way that would keep our civilization alive? I mean is it practically possible to obtain technology that would enable us to travel to nearby habitable earth-like planets to keep our civilization alive?
For example, consider this design for NASA’s Star Trek-style Space Ship, the IXS Enterprise. Do you know any good site which goes into detail on both latest practical and theoretical development of this interstellar flight?
The link you posted refers to a design which would supposedly make use of faster than light travel. After reading this, I immediately stopped reading, since this is not possible. I don't care if it's a NASA-affiliated person who says this, it is simply misguided!
Before continuing, I should draw your attention to the fact that I will not be discussing the Alcubierre drive, a highly speculative method of achieving very fast space travel which I personally think is more based on wishful thinking than anything else. Instead, I will stick to methods of travel that to not depend on manipulating spacetime in an exotic way.
A different question is whether it might be possible without using faster than light travel, although the answer is almost certainly still no. From this wikipedia article we learn that the closest possibly habitable planet would be around 12 light years away. This means that, traveling at $c=299792.458\ \frac{\text{km}}{\text{s}}$ one would take about twelve years to get there.
Let's assume a best-case scenario, where this planet turns out to be absolutely perfect - with minimal issues pertaining to adjustment to a different atmosphere etc. - while we also ignore any logistical issues like building enough spacecrafts to transport a significant number of humans (note that these issues alone will probably already make moving to a different planet an infeasible plan in a realistic scenario!). Let us just focus on the travel time.
From some further wikipedia research (notably here and here) we can conclude that our fastest (unmanned!) spacecrafts currently have a speed around $20\ \frac{\text{km}}{\text{s}}$. Let us, for the sake of discussion, assume that we can significantly improve on this "in the near future". Say some great technological breakthroughs occur that allow us to improve the speed of our spacecrafts by over an order of magnitude(!) to about $v_\text{sc}=300 \frac{\text{km}}{\text{s}}$ - this number is chosen somewhat arbitrarily; it makes comparing to the speed of light easy. Then, our estimate time of arrival would be
$$\text{ETA}\sim 12\ \text{year}* \frac{c}{v_{\text{sc}}}\approx 12*10^3\ \text{years}$$
As you see, we might experience some trouble here, too ;) In short, I think it is safe to say that we will not be able to move to a different planet in the foreseeable future.
It's easy after we rebuild ourselves first. We need to replace biological brains by digital brains. The problem we now have is that if we travel, we carry a biological machine that is much more advanced than the most powerful supercomputer we can build today, physically to the point of destination. This biological machine must always be kept at the right temperature, it needs to be at the right atmospheric pressure etc. etc.. Obviously, it would be far more practical to have a digital version of these biological machines and then simply upload the data to a machine at the point of destination.
The question is then how to bring machines with the necessary infrastructure to the next inhabitable planet? This may be done using nanotechnological means, you can imagine that only need to send a microscopically small device which only needs a small rocket to a distant planet and then the device will grow all by itself and we'll have the machines ready to receive us via radio transmissions (or perhaps laser transmissions which have a far lager bandwidth).
Another possibility is to send messages to distant civilizations. While two way communications take a long time, to upload ourselves to another civilization only requires one way communications. All we need to do is to repeatedly transmit our messsages that catch the attention of some distant civilization followed by the data containing all the information needed to be able to run our brains. It may be that our civilization will be long destroyed when the messages are still on their way. If millions of years later we are received by a civilization in the Andromeda galaxy and they manage to run our programs, we'll not perceive this time lag. To us it would feel like we've arrived there at an instant.
As @Danu said, the transit time is a massive problem. One of the comments to his answers mentioned that keeping the passengers alive might be difficult too. You need
These factors are important if you're going to be in space for more than about a year, so manned visits to neighbouring solar systems might be impossible even if your spaceship is a couple of orders of magnitude(!) faster than @Danu's.
