In this question I am not considering the sun, but only other stars.

Stars are big, but they are also far away. https://en.wikipedia.org/wiki/Star

The nearest star to the Earth, apart from the Sun, is Proxima Centauri, which is 39.9 trillion kilometres, or 4.2 light-years away.

In pictures like this:

I believe that the size of the spots is due to overflow of charge in the CCD, and that the four rays is due to optical artifacts in the telescope. (Please correct me, if I am wrong)

What is the largest space angle of any star? What is the best angular resolution attainable by any current telescope? Is there any theoretical limit to angular resolution attainable by telescopes?

So is it possible to take a picture of a star, where the star is not just a single point or pixel?

Update: The spot size is not due to charge overflow - see answer by Chris White.