The diet of the ordinary Greek in classical times was largely vegetarian--vegetables, fresh cheese, oatmeal,
and meal cakes, and meat rarely.

(A) and meat rarely
(B) and meat was rare
(C) with meat as rare
(D) meat a rarity
(E) with meat as a rarity

A and B: The diet of the ordinary Greek in classical times was largely vegetarian--vegetables, fresh cheese...and meat.
Here, the usage of and meat implies that meat is a VEGETARIAN food.
This meaning is nonsensical.
Eliminate A and B.

C: meat as rare
meat as rare AS WHAT?
Also, the usage of rare could imply RARE MEAT -- in other words, meat that is barely cooked.
Not the intended meaning.
Eliminate C.

An absolute phrase is a modifier composed of COMMA + NOUN + OTHER WORDS that serves to modify the entire preceding clause.
In an absolute phrase, the noun after the comma must refer to the PRECEDING SUBJECT.
SC108 in the OG for Verbal:
Sound can travel through water for enormous distances, its acoustic energy prevented from dissipating.
Here, the portion in red is an absolute phrase in which its acoustic energy (the noun after the comma) refers to sound (the preceding subject).

D: The diet of the ordinary Greek in classical times was largely vegetarian, meat a rarity.
Here, the portion in red seems to be an absolute phrase, but meat (the noun after the comma) does not refer to the diet (the preceding subject).
Eliminate D.

The correct answer is E.

Generally, COMMA + with at the end of a sentence serves as an ADVERB modifying the VERB in the preceding clause.
OA: The diet of the ordinary Greek in classical times WAS largely vegetarian -- vegetables, fresh cheese, oatmeal, and meal cakes, WITH meat as a rarity.
Here, the portion in red is an adverb modifying was -- the verb in the preceding clause -- expressing how the diet WAS largely vegetarian.
HOW was the diet largely vegetarian?
WITH meat as a rarity.

Hi GMATGuruNY,
Apart from what you mentioned above, please let me know whether the followings are potential ERRORs -

Option A: meat rarely -- rarely is an ADVERB, so it can't modify a NOUN (re MEAT).

Option C: with meat as rare -- rare is an ADJECTIVE, so this PHRASE becomes ungrammatical/nonsensical.

RBBmba@2014 wrote:Hi GMATGuruNY,
Apart from what you mentioned above, please let me know whether the followings are potential ERRORs -

Option A: meat rarely -- rarely is an ADVERB, so it can't modify a NOUN (re MEAT).

Correct.

Option C: meat as rare -- rare is an ADJECTIVE, so this PHRASE becomes ungrammatical/nonsensical.

On its own, NOUN + as + ADJECTIVE is not grammatically viable.
But this phrase can serve to INTRODUCE a correct structure, as in the following:
an event as rare as a solar eclipse