Indefinite articles used with plural nouns: It was AN amazing TWO DAYS

Solution 1:

A few old grammar rules

A great many, a good many, a few.—These are very incorrect and bad phrazes; and the singular article can never be properly used with a plural noun.

Since Few words on Many Subjects was published in 1831, English has seen quite a few changes. I don't know whether this rule was enforced at schools, but I did find another example lambasting the use of the indefinite article before many in front of a plural noun.

There is an extensive and growing error in the use of the adjectives good and many. It is not correct to say “a good many apples,” tho we may say “many good apples;” for, omitting the adjective good, we can not say “a many apples.” Neither is it correct to say “a great many persons;” for, “a great persons,” or “a many persons,” would be improper. It would be better to say, many apples; many persons, and omit the good and great. We do not hear of “a bad many,” or a “small many.” Why then say “a good or great many?” “The rushing of many waters;” “the influence of many minds,” are much more expressive than to add the words great and good.

^{Source: A grammar of the English language: Explained According to the Principles of Truth and Common Sense ... (1839)}

a/an + adjective + number + plural noun

The fact that native speakers were using the article "a" in front of adjectives and plural nouns in the 19th century, proves there is nothing new under the sun. Today the following sentences are perfectly grammatical.

1. He weighed a whopping twenty-five stones (350 lbs)
2. We spent a wonderful/fantastic/memorable three weeks in Greece
3. He had collected a good many books
4. She waited a full three minutes before speaking
5. It had been an exciting two years for Alice
6. Ted had an exhausting two days in Denver.
7. It costs a mere twenty dollars.

In sentence 3, “a great number of” could substitute “a good many”.

(a) He weighed twenty-five stones.
(b) We spent three weeks in Greece.
(c) He had collected many books.
(d) She waited three minutes before speaking
(e) Alice had experienced two exciting years.
(f) Ted had two exhausting days.
(g) It (only) costs twenty dollars

The sentences are only grammatical without the "a" and its "adjective"; take away only one of the two components, and the sentences become ungrammatical. The indefinite article modifies the adjective with the number. There has to be a number attached to the adjective in order for the sentence to be grammatical.

“She waited a full minutes before speaking” (NO)
“She waited a three minutes before speaking.” (NO)
“She waited a minutes before speaking” (NO)

The noun phrases take a plural noun and a plural verb after the singular a great /good many; or a/an + adjective + number; e.g.

A great many people in this country are worried about law-and-order
An astronomical 300,000 tons of apples were destroyed.
… an amazing 250,000 new neurones are added every minute.

In a great (or) good many people, “great” and “good” act like the adverb very, or really, they intensify the adjective many; i.e. “very many people” and “really a lot of people”.

The determiner many and a good many are listed in all the dictionaries I checked, but they offer no insights as to why this construction is acceptable.

many n. (used with a pl. verb)
1. The majority of the people; the masses: "The many fail, the one succeeds" (Tennyson).
2. A large indefinite number: A good many of the workers had the flu.
^{American Heritage® Dictionary of the English Language}

man•y
3. a large or considerable number of persons or things: A good many of the beggars were blind.
^{Random House Kernerman Webster's College Dictionary, © 2010}

Rogermue in the comments, suggests that a great many is derived from the German noun Menge a word meaning "multitude".

An article in Language Log has this to say on this particular construction

Funky a

A couple of days ago, I took Roy Peter Clark to task for claiming that phrases like "a million dollars" show that the indefinite article a can be used with a plural head ("Slippery glamour", 7/4/2008). I observed that the structure is clearly [[a million] dollars], not [a [million dollars]]; that expressions like "a million" are just numbers, fitting into the normal syntactic slot where numbers go; and that million in this case is morphosyntactically singular.

In the comments, Russell Lee-Goldman pointed out that
There are, however, a few cases where it really looks like "a" is acting funky:

– He was there for a good seven years.
– An additional three people are required.
– A mere four nations recognize that standard.
– She collected an amazing and heretofore unprecedented forty million dollars.

[..examples taken from the web...]

But these examples seem to me to represent a generalization of the phenomenon on display in phrases like "a million dollars": English number-expressions have inherited from their partitive history a limited ability to act like singular noun phrases.
However, I'll admit that the constituent structure doesn't feel like

[ [a <modifier> <number>] <noun>]

but rather feels like

[ [a <modifier>] [<number> <noun>] ]

— for what little those feelings are worth

The article ends with an update and suggests reading two studies. The paper, A SINGULAR PLURAL, by Tania Ionin & Ora Matushansky, I believe would interest the OP a great deal.

Solution 2:

OK. I think the singular article is used because in each of your examples the plural nouns make up a single unit of time. The same structure would be used with other units of measurement:

a weighty three bags of coal

a full ten bottles

an arduous eighty kilometres

In such case, the unit of measurement is the noun phrase which includes the amount: "three months"; "ten bottles"... so the adjective comes, naturally, before that noun phrase.

Without the article, there is no single unit, so the quantities refer to multiple units, which are therefore plural nouns and take their adjectives just before themselves.

Does that make sense?