How is the size of stars measured?

To measure the size of a star, it's simply a matter of angular resolution (the smallest detail that can be seen). The angular resolution of a telescope is given by the following equation:

θ≈1.22λ/DD

Where λλ is the wavelength of light you are observing with, and DD is the diameter of your telescope.

However, if you combine the light from two telescopes that are far apart (which is not at all easy to do!), then DD is the distance between the two mirrors. This is called interferometry, and is a widely used technique in astronomy.