Greek Gods were not like the 'gods' of the West. Zeus was unlike the Jewish, Christian or Muslim God- he was unlike them both in having colleagues, or either sex, in having a geneaology and a succession, powers limited to a region of the world and his own set of special tools (the thunderbolts). When we turn to look at the Gods of Greece we look at entities which are not the same as our own deities- and thinking about them according to the categorisation that we have developed for whatever God we believe in or oppose may not be useful. Ancient and particularly archaiac Greek polytheism- which is really what we are talking about here- is a very different beast to modern religion.
It is a different beast in part, as Jean Pierre Vernant argues, because we have two related attitudes to the body which the Greeks did not have. The Archaic Greeks did not believe that the body and soul were distinct entities and they did not believe that the body was something separate that one could study. For them the body was the person: this breeds that remarkable Greek attitude (remarkable to anyone brought up in a culture based on the separation of body and soul) that the external beauty of the body reflects moral virtue, that ugliness is in some sense a moral hazard. If the Greeks did not accept a fundamental division between body and soul, neither did they accept a fundamental division between nature and supernature. Zeus and Aphrodite slept with mortals, Ares battled with them on the planes of Troy and a mortal (Paris) was even appointed to arbiter between Aphrodite, Athena and Hera when they were in discord. We should not be surprised to find both these ideas- that the human being was indivisible and that the Gods and humans lay on a continuum influencing what the Greeks understood by a divine body.
The Greek conception of a human body was that it was perishable- every Menelaus turning eventually into Nestor, every Helen into Aethra. Throughout life the body needed revitalising by sleep (Hupnos) and eventually of course the term of the body ran out and the being became subject to the twin brother of Hupnos, Thanatos or Death. The Gods though were not mortal but immortal- in that sense they had 'super' bodies. Aphrodite's beauty was distinct from Helen in that it would not fade. The predicate of immortality was fastened to these 'super' bodies- but also they were exalted because of the nature of their beauty- as the Homeric Hymn comments on the Ionians at the island of Delos 'an unexpected guest would think them immortal and free from old age for he would see grace in all of them'. Notice that immortality comes with extraordinary grace or beauty or indeed strength (hence Apollo can kick away the fortifications of the Argives at Troy like sandcastles on a beech despite their great efforts to construct them). Beauty, strength and grace are all opposites to decay, to old age and ultimately to death- hence they are predicates or permanence: Keats was right in the Ode to the Grecian Urn that one of the great issues in Greece was of the immortality of beauty and that if beauty was immortal it was divine and hence true.
A last thing that Vernant notes that I found very interesting is the way that Greek Gods intervene in the world. Whether it is Odysseus meeting Nausicaa or similarly meeting Telemachus in Ithaca, Athena gives him added lustre. Interestingly that is accompanied by Odysseus getting dressed- and that moves us on to another interesting aspect of the Greek attitude to the body. They saw the aspects of the body whether it be the armour of Achilles or the thunderbolt of Zeus as parts of that body. When Pandora was created, she was created with all the implements of seduction, jewelry, dress and necklace. This adds another metaphor about divine intervention in the world of the human- the Gods putting on (like Prometheus did to Ajax) the hero courage and other attitudes. In a sense, as the outward persona and the inward persona are the same and the body shapes (in our language) the mind, dress becomes a signifier of a change within a person: a God can achieve this but so could a person by dressing themselves.
What Vernant gets here, and I think it is interesting, is a different mentality and way of seeing the world based on a series of very simple assumptions about the world- in which the Greeks and we differ. I want to finish on a caution though- I have relied on his research- and yet there is a reason to be sceptical. It is incredibly difficult to get to a different mentality and there is the potential for evidence to be used and chosen selectively: Vernant seems to use enough evidence to suggest he is not doing this- and even if he has constructed a whole out of a series of shards which denote something else, I think what he has constructed is interesting. It is a different way of viewing the world which leads to several basic assumptions that we all make being discarded, part of the reason to think about history is to realise how strange the past was (History as C.S. Lewis suggested is a traveller wondering through distinct, strange and wonderful territory)- in that sense Vernant succeeds in providing us with a strange and novel vision of the world. The Greek view of the divine body and the human body and its relationship was a fascinating one- and deserves to be understood partly because it is so counter intuitive and strange to those reared in a world that assumes a mind body, nature supernature duality.