Is this sentence correct?

No.

“Me” should be “I”, because “I” is the subject of that clause, which written directly would be “I was in the same English class”.

An alternative wording could be: “People who were in the same English class that I was.”

It’s also wrong because there should be punctuation after “with”, such as a period.

So I’d correct this to proper English as:

“The same people I went to school with. People who were in the same English class as I, who turned out to be illiterate…”.

Or you could use a dash:

“The same people I went to school with – people who were in the same English class as I, who turned out to be illiterate…”.

That was a messed up sentance.

I can not tell from that sentence, if it was you who turned out illiterate, or the people you went to school with.

The comma should follow “with” not “people”. In the same English class with or alongside ME. I think if you’re going to use “as”, the sentence should end in “I”. I can’t tell you why. But I think the test is that you can’t say… the same class as “me did” “or me was”. “Myself” is probably better than “I”or “me”.

My English classmates turned out illiterate.

As originally written the sentence is unclear, verbose, and bad grammatically.

Much better and more concise would be:

People who were in the same English class as I turned out to be illiterate.

What makes the sentence in the original post awkward is that is has the words “people” and “same” twice.

I did a Web search for “as I” versus “as me”. The official rule is that the word “as” is a conjunction and that “as I” is a shorthand form of “as I am”. I prefer to think of “as” as a preposition in this case, so that “as me” would be acceptable. If you use “as I” then you also have to say “as he” and “as she”.

^You run into similar problems with the word “than”. Cf. the following three sentences:

1. He is taller than me.
2. He is taller than I am.
3. He is taller than I. (?)

Some English speakers will say that sentences 1 and 2 sound fine but #3 sounds odd (that’s how it is to me. I would never say Sentence #3). Grammar guides will say that #1 is incorrect and #2 and #3 are correct if “than” is a conjunction, but #1 will be correct if “than” is a preposition. Ultimately these words are conjunctions in origin, but they are used like prepositions in some cases. The subject forms of the pronouns (“I’, “he”, she”, etc.) are most acceptable to use when there is a verb for them to be the subject of. An “implied verb” often isn’t though, thus the object forms are used.

One thing I’ve noticed about the object forms of pronouns in English is that they’re not so much object forms as much as they are not-subject forms, used in all cases where they are not the subject of a stated verb, such as in phrases like “Goodness me!” or “it’s me”. In neither of these is “me” being used as an object. “Me” is even used as a topic marker, e.g. “Me, I’m more the stay-at-home type”.

Frankly, as long as you get your meaning across, it does not matter.
Just as dictionaries are just a documentation of common usage, so are rules of grammar just observations about language in current use.

@Zaku This is just plain wrong: ”....The same people I went to school with. People who….”
That period does not belong after the word with! Periods end a sentence. That first bunch of eight words is not a sentence.

As for I vs me, I agree that “I” is the correct pronoun there.
Using ‘me’ in that spot… bad grammar that is generally used because people don’t know grammar doesn’t make it correct.

It’s not incorrect if “as” is a preposition though. To me “same class as I” without “was” sounds pedantic.

There was so much wrong with that sentence as written that arguing over “I” or “me” is futile. “I” is what is correct grammatically but “me” is more and more common and sounds ok too. As I said in my first comment, the sentence is a hot mess of redundancies, incomplete phrases and incorrect punctuation.

I’m late to this, but I didn’t see this mentioned outright already so I thought I’d add: The sentence is not a sentence (not unless the ellipsis is cutting off the rest of it).

“The same people I went to school with” = subject, noun phrase

“people” = extra word

“who were in the same English class as me,” = subordinate clause (or something—I don’t remember all the terms for grammatical parts of sentences—but in any case it starts with “who” and that attaches it to the description of the noun.)

“who turned out to be illiterate…” = another subordinate clause (it also starts with “who”)

So as far as the structure goes, it’s missing the verb/predicate part of the sentence.

I’ll make up a predicate phrase to better show what I mean: “The same people I went to school with, who were in the same English class as me, who turned out to be illiterate, still managed to complete the reading assignment that I forgot to do.”

(It obviously changes the meaning of the sentence, the sentence isn’t be “good” this way, but I believe it is now a complete sentence. I assume the original sentence’s intended meaning is what @janbb revised it to say.)

… As for the other part of the question, I don’t personally care whether it’s “as me” or “as I,” I think both work just fine.