I took a bunch of teenagers shooting last month, and most of them had never picked up a gun before. There were three boys that outshone the others as being much better shots straight away. And funny enough, they were not the boys I would have predicted. So, yes, people can have a knack for it.

That being said, better than average depends on what population of people you're talking about. If you're including all non-experienced shooters, then absolutely, with a little careful practice, most people can become decent shots in a pretty short amount of time.