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A common misunderstanding of special relativity is that you can't go faster than light. In actuality, it depends how you want to define things. If you divide the distance from earth to wherever by the time it takes the astronauts according to them, then this could be greater than c. A more consistent definition of velocity requires that the distance and time be measured in the same frame. It is this which cannot ever exceed c (in SR).
I think this type of thing is very doable, but it leads to problems if the travelers ever want to return to earth (everyone there will have aged much more). There are also the more mundane issues of shielding the spacecraft from interstellar junk and blueshifted radiation at those speeds.
As mentioned by others, there are possible loopholes using general relativity. The Alcubierre warp drive does not seem possible right now though due to its negative energy requirements, causality problems, etc. It also requires a ridiculous amount of energy. The best version would require a few suns worth of fuel. And nonexistent negative energy suns at that.
As others have said, our understanding of physics could change dramatically on this issue in the future. We'll see...
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