The same way you survived the acceleration up to that speed; spread it out over time. Astronauts on the space station are going over 17,000 mph relative to the ground, and getting up to that speed they have to pull a couple of g's in a rocket, which is tough but doable for a few minutes. On their way back to Earth the atmosphere slows them down a little more gradually and they only experience 1-1.7 g's.

And not that we would ever want to stop relative to these things, but just sitting at your desk you're moving incredibly fast through the universe. The Earth is moving ~67,000 mph around the sun. And the sun is moving ~490,000 mph around the center of the galaxy.