The only way to describe the electron radius that I found in literature is the "classical electron radius". Is it possible to experimentally measure this? Is there a better way to describe the electron size (and in general, the size of any other elementary particle)?
Apart from a few special cases the way we measure the size of a particle is to scatter other particles off it. The Hyperphysics site has a nice introduction to this - the article talks about measuring the size of nuclei by scattering, but this applies to any object. The scattering from a point like particle is different to a particle with a finite radius, and from the difference in the scattering patterns we can measure the radius.
The measurement can only be done if the de Broglie wavelength of the particles you are scattering is smaller than the size of your target. Since $\lambda = h/p$ this means really small objects can only be measured by using very high momentum and therefore very high energy. The reason I mention this is that at the energies we can currently reach (8TeV at the LHC) all the fundamental particles appear to be pointlike i.e. their radius is zero. Composite particles like protons and mesons (both made from quarks) have a non-zero radius but the quarks themselves appear to be point like.
So the electron radius is zero - as far as we can tell at the moment. Though if string theory is the correct fundamental description of particles then electrons and all fundamental particles have a radius of the order of a Planck length.