This has nothing to do with acceleration. The idea is that you bend space-time so that you experience movement of a relatively short distance but actually travel hundreds of light years. In order to do so you have to alter space-time as we know it. Just a few weeks ago, we found we don't even know what happens at the edge of our own solar system. How can you think that bending space and time will act the same way as driving faster in a car? Your experience could be movement in the range of a few KM, but in the end you could be light years away. The problem (and the reason you will surely perish if you try it) is that you need to change the way space and time fit. Go ahead and try it, just be a few hundred light years away first so you don't ruin our perfectly functional solar system.
