I do realize that the phrase "to be man enough" is an idiom. But I wonder what is the grammatical/syntactic role that man plays in it. Is it an uncountable noun? An adjective? An adverb? Or perhaps because the phrase is an idiom, the question makes no sense?
Consider an alternative sentence:
He is sufficiently manly to achieve his goals.
Here manly is an adjective and sufficiently manly is an adjectival phrase. In your sentence, man enough is a colloquial substitute..........so it is also adjectival.
Because of its colloquial nature, I would only catagorize the phrase and not break it down further.
Man is a noun in constructions such as 'to be man enough'.
To be more specific, man is a predicate noun, or a noun used in the predicate of a sentence. The Oxford English Dictionary (OED), under the definition of enough, gives an example of this exact phrase, from, coincidentally, another dictionary, the New English Dictionary (N.E.D.).
1891 N.E.D. He was not man enough to confess the truth.
The description in the OED of enough in this usage is
quasi-adjective, qualifying a noun used as a predicate
Note the term quasi-adjective describes enough, not man. Other OED examples that fall under this same entry for enough are
1711 The Butler has been Fool enough to be seduced by them.
1878 Looking at a spot into which she was not climber enough to venture.
Fool and climber are nouns.
In additon, the first definition of enough in the OED says:
- in concord with noun, expressed or implied:
and as examples for with noun in singular it gives uses from Old English (prior to ~1100) through Middle English to Modern English, and they include
1518 With payne and trauayle anough, we made toward the Cowrte. (With pain and travail enough, we made toward the Court--my updating of this sentence)
approx1616 There's wood enough within. (Shakespeare, The Tempest)
1766 He had not resolution enough to give any man pain by a denial.
1816 That thought is happiness Enough for me.
Webster's 1828 Dictionary notes that
This word, in vulgar [common] language, is sometimes placed before its noun, like most other adjectives. But in elegant discourse or composition, it always follows the noun, to which it refers; as, bread enough; money enough.
Edited to add:
Another modifier that goes after the noun is alive. And here is an example:
As for the scientist, he has absolutely no use for me so long as I am man alive.
Another way of phrasing it is, "He is enough of a man." Then "enough" is an adjective, but "man" is something of a "qualifier," even though it is technically a noun. That may be where the confusion is coming from.