As you move faster, according to special relativity, time actually slows down. Let's take the 500 light year number mentioned by Hobbes answer, and work with that. So let's assume you have a 60 year lifetime. That would require a time dilation factor of about 500/60, or 8.333. I found an online calculator that states you would have to go 0.99277c to achieve that dilation factor. For a smaller lifetime, the speed would have to increase, but it is possible, if you can achieve near light speed somehow, that you could reach any distance. Of course, anyone on Earth would have the full 500 years pass (Well, 500/speed, but close enough)
Practically speaking, achieving such speeds is virtually impossible, and then there's the chance of an impact, which would be quite serious. Still, it is possible.