Aside from the problem discussed in the article of squishing people by accelerating them at hundreds of g, there's a more fundamental problem with a "launch X into orbit by shooting it very fast" approach: Trajectory. An object in freefall (which your ballistics are once they leave your gun / catapult / linear accelerator / etc) with a bounded trajectory (i.e., doesn't escape Earth's gravity entirely) travels in an elliptical path. If one point on that path is on the earth's surface... well, there's 3 more points on the earth's surface, too.
In order to reach orbit, you MUST add angular momentum. And until we have a space elevator, that means rockets.