The first reading is untenable, especially in light of verses 53-56--but then, we knew that. The second reading is really somewhat far-fetched, as the plain grammatical sense of the text is that *the flesh* profits nothing. But this flesh can't be the Divine Body, as we know, and this it must be something else--as EA noted, our sinful nature (the passions). (This is also the technical meaning of the term in the Pauline epistles.)
Here is St John Chrysostom's commentary on the passage:
His meaning is, “Ye must hear spiritually what relateth to Me, for he who heareth carnally is
not profited, nor gathereth any advantage.” It was carnal to question how He came down from
heaven, to deem that He was the son of Joseph, to ask, “How can he give us His flesh to eat?” All
this was carnal, when they ought to have understood the matter in a mystical and spiritual sense.
“But,” saith some one, “how could they understand what the ‘eating flesh’ might mean?” Then it
was their duty to wait for the proper time and enquire, and not to abandon Him.
“The words that I speak unto you, they are spirit and they are life.”
That is, they are divine and spiritual, have nothing carnal about them, are not subject to the
laws of physical consequence, but are free from any such necessity, are even set above the laws
appointed for this world, and have also another and a different meaning. Now as in this passage
He said “spirit,” instead of “spiritual,” so when He speaketh of “flesh,” He meant not “carnal
things,” but “carnally hearing,” and alluding at the same time to them, because they ever desired
carnal things when they ought to have desired spiritual. For if a man receives them carnally, he
profits nothing. “What then, is not His flesh, flesh?” Most certainly. “How then saith He, that the
flesh profiteth nothing?” He speaketh not of His own flesh, (God forbid!) but of those who received
His words in a carnal manner. But what is “understanding carnally”? It is looking merely to what
is before our eyes, without imagining anything beyond. This is understanding carnally. But we
must not judge thus by sight, but must look into all mysteries with the eyes within. This is seeing
spiritually. He that eateth not His flesh, and drinketh not His blood, hath no life in him. How then
doth “the flesh profit nothing,” if without it we cannot live? Seest thou that the words, “the flesh
profiteth nothing,” are spoken not of His own flesh, but of carnal hearing?