There is a lot of heat that will come from the guys with accurized, heavy-bbl AR's. They are correct, these things are accurate. On the other side, benchresters (The MOST accurate kind of shooting) all use bolts. I'm going to point out, the question is about rifle actions, not shooter skill. We leave the shooter's skills out, here.
Having said the above, let's get practical. You can achieve fine accuracy with a high-quality autoloader. The military shooting teams demonstrate this all the time. You can achieve the same or better accuracy with a high-quality bolt action. Many non-military target shooters still use bolts, to excellent effect. Both work for hunting, and for any other civilian activity, equally well.
(For military use, in modern combat, there is no question but that the autoloader is preferred—but the philosophy there is "spray and pray," at which point we are no longer talking about accuracy.)
The major difference is in the cost. Autoloader guys claim the cost is equal, but then they go out and spend close to or over a grand on a gun that then has to be accurized. You can get a really nice bolt gun for a few hundred, accurize it as much as you think necessary, and have change left from a grand.
If you just buy a gun at the store and accept the factory level of accuracy, you will always pay more for the autoloader. It's a more complicated piece of machinery.
Lever guns, pumps, break-actions, double rifles, falling/rolling block single shots, and so on, do not have the same inherent accuracy. To my knowledge, there is no way to fix this.
There are other considerations in the choice of an action, for example, autoloaders aren't available in nearly as many different calibers as are bolts, but the question here is one of pure accuracy, which I hope I've answered.
