Doesn't the standard model suppose that ALL elementary particles (e.g. electron, neutrino, quarks) have zero size; i.e. they are true pointl-ike particles?
Yep it certainly is. The concept is that an electron is a point entity, but then it creates infinities which do not play nice with the universe. "Thing" being ill defined I'd have to go with that which the Planck Length represents, the smallest distance where space and time have any real meaning as we understand it. Something smaller might just "fall through the cracks" and be gone.
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