Chromatic aberration occurs because the lens is does not always foucs all the different wavelengths onto the image sensor at the same point. The results are usually seen as a purple or green fringe on the image. Shooting wide-open or including high-contrast areas in the frame increases the likelihood of seeing chromatic aberrations. Stop down a few stops to keep aberrations to a minimum.
In the 200% cropped picture below, you will notice that the DA* 300mm exhibits very slight purple fringing on the palm tree.
If the DA* 300mm has distortion, then it can barely be discerned in real life.
The DA* 300mm does exhibit minimal chromatic aberration, but it is so slight that in most situations it won't show itself. Barrel distortion is negligible.