Aren't you being too hard on yourself ? I suspect it DOES make sense. There are plenty of hits for the phrase έχει ζωή on Google (stuff like Έχει ζωή η Ευρωζώνα; which I assume means "Is there life in the Eurozone?"
So could it not mean "He has life in him" or "He's alive" ?
(But not "He's a life" – please chop that one off the list).
The hints are just hints, not directives, and they are intended to point, not to guarantee. They are directed towards the meanings of individual words or short phrases, but not so much to whole sentences, where greater context can shift meanings to a degree.
The sense I get from comments made earlier in this discussion is that in general, Greek use of the indefinite article is applied on somewhat different occasions than it is in English, so when translating into English we must apply the English rules, and when translating into Greek, the Greek rules - which do not always match. The Greek meaning of the sentence here, without the article, seems to match better the English meaning of a sentence that contains the article, and that is a key aspect of knowing Greek usage better. Sometimes, in translation, it's not so much "close, but no cigar"; rather, it's "close but no copy".
"η ζωή" would be "The life" and not a life, but "μία ζωή" is certainly an interesting problem here. My understanding of this problem is that many languages don't use the indefinite article(s) as strictly as English does, only when they really want to express that we are talking about ONE (and only one) of the mentioned category. It seems that Greek goes this way, which is certainly a relief for me whose native language (Hungarian) also works that way (Imagine how many times I got wrong points in the Esperanto course because I forgot about the fact that English has indefinite articles and uses them almost all the time).
"He has a life" – OK, why not? The explanations given below are certainly amusing! But the answer given on the actual page is "He's a life", which is silly (and borderline ungrammatical). No-one, unless they had had an ouzo too many, would make that contraction. It could only mean "he is a life", not "he has a life".
The hint "he's a life" presents the same kind of problem that occurs in other places as well. Even though the verb means "he has" the program seems to be generating "he's" as a contraction of "he has." There's a discussion of the technical problem under "Αυτός έχει μία εφημερίδα."