This is where you're wrong and it also seems that whoever typed that quote is also wrong/misguided. It's a common misconception to apply the E=mc^2 formula to photons and therefore think they have mass.
no, he wasnt wrong.
If this pair of electron and positron were to collide each other and turn themselves into photons, their previous rest mass did not just simply vanish but became mass-energy of the photons.this is basic. niether mass nor energy is ever destroyed.
again, if something exists it takes up space. if something takes up space, it has mass.
i will suggest to you though.. if you consider the idea of energy as not existing at all in physical terms.. but only in a supernatural "ghostly" manner.. like a spirit which can be seen but is transparant, permeable, and not physical... then its possible for it to exist and not have any physical mass..