As far as I know, 20 or 25 km/s is the maximum velocity achieved by practical spacecrafts. If such a Sci-fi craft exists (as in Wall-E - where the craft cruises through the debris), we have a lot of choices. Physics is not necessarily needed. Sci-Fi is very good.
Spacecrafts (I mean, those used in space-travel) of the past and present are mostly guided by a bunch of astro guys on ground. If they were able to design the crafts and guide them through the asteroid belt (like the Voyager twins) by using some transmitter-receiver mechanism as in RADAR or SONAR or a Computer-aided Guidance system, then we could think of the future easily.
First, we'll be in an age of space-travel, where we'll have the farthest seeing telescopes (best ever ones than Hubble or James), and we'll have all exact locations and events (that may happen during travel, like future impacts on our space ship, etc.). So, guiding the craft wouldn't be so difficult like making some Brownian motions like the molecules do, or even land in several asteroids resting some time, like that.
If it simply by designing the spacecraft to have huge acceleration, wouldn't that mean lots of constraints on the material the spacecraft can be made of?
Not only the material because, the material is required only to provide resistance to the impact-able asteroids or when you're planning to dash through all the objects on your path. If the craft is planned to make sudden turns on its way, it should also have opposite thrusters with more or less the same power. If I'm allowed to talk still, I'll go into Sci-Fi.