What is the size of atomic and subatomic particles, like proton, neutron, photon etc? Is it defined based on some quantum characterics as de Broglie wavelength or Compton wavelength?
The size is determined by the scattering cross section.
Classically one could measure the size of a leaf ( for example) by measuring its shadow from a light source, if the geometrical distances were known.
For elementary particles it is more complicated because the projectile and the target interact quantum mechanically but still we can get an estimate of the size of a proton when seen by photons, or by muons or by neutrinos by using the appropriate parameters of the interactions involved.
We get "average sizes" because nothing is sharp in quantum mechanics and it depends on the momentum transfer during the interaction in a non trivial way, but we do get a good measurement of its size, and distribution of quarks within the protons, neutrons and pions by measuring form factors.
Neutrinos, photons and electrons are deduced/assumed to be point particles of dimension 0, in these scattering experiments.
An alternative to the (correct) method that anna suggests is to measure the momentum distribution of the sub components and apply a Fourier transform. As a matter of experimental difficulty these kinds of studies usually come later then direct cross-section measurements, but they provide a different kind of data and can throw a new light on the particles probed.