The size required to do this is a function of the materials that make up the object. There are a number of articles on the internet that explain what it takes. The bottom line is, the stronger the material, the more of it it takes to make the object spherical. Imagine if there was a mountain 500 miles high on Earth. It would require tremendous strength to stand up to the pressure from the rest of the planet trying to pull it back down, and in fact, this wouldn't work for very long. Basically, the mountain will flatten itself out, spreading it's pressure over a larger area.
Owing to the above, every planet (Or asteroid, etc) has a maximum height for mountains. Further physics has an article that explains this, for a purely vertical block. For a triangular shaped block, the distance is about twice as much. Mount Everest is close to the maximum theoretical height for a mountain on Earth, although slightly higher is possible. Smaller planets allow for higher mountains. Eventually, one reaches a point where it doesn't matter how high a mountain is, it won't make a difference. That is the point where an object is considered near spherical.
It turns out that the size where one can't have mountains larger than the object turns out to be size at which an object will be spherical. Ceres, for instance, could theoretically have a mountain 177 km high, which is well below its size. Vesta, on the other hand, could support a mountain 234 km high, which is basically the same size as its radius of 260 km. Thus, Vesta is nearly spherical, but can have some significant deviations, as can be observed in the images taken by Dawn.