Greetings Phlogistician:
If I may be so bold as to offer something here have you ever read about any Orion Project propulsion research that was conducted back in the late 50's (I believe)?
You should do your research. Project Orion has of course been discussed, and debunked, right here at SF:
http://www.sciforums.com/showthread.php?t=40942
We currently have the technology that could be used to accelerate an enormous amount of mass to a substantial percentage of the speed of light.
No we don't.
Also, it was announced yesterday that the Kepler space probe has yielded data that shows that there are at least 100 BILLION planets in the Milky Way, 1500 of which are within 50 Light Years of Earth. With all that in consideration interstellar travel is far from impossible.
We know there are lots of stars. So what? We can't detect if any have atmospheres, or life yet, so cannot pick a candidate to visit, even if we had the capability, which we just do not. Interstellar travel IS IMPOSSIBLE at present, we can't even get back to the Moon, let alone another planet in our own solar system!