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Suppose that Eve said (in spoken English)

An apple a day keeps the doctor away, unquote. Also isn't Eve such a
great person? Like my mom always said quote Eve is the best person ever,
much better than that Alice person

Now let us say that Bob wants to tell Alice what Eve said. Naively, he might say

Eve said the weirdest thing the other day. She said quote An apple a
day keeps the doctor away, unquote. Also isn't Eve such a great
person? Like my mom always said quote Eve is the best person ever,
much better than that Alice person unquote. What do you suppose it
means?

The problem with saying this is that Alice would interpret this as

Eve said the weirdest thing the other day. She said "An apple a day
keeps the doctor away." Also isn't Eve such a great person? Like my
mom always said "Eve is the best person ever, much better than that
Alice person." What do you suppose it means?

Whereas Bob actually meant

Eve said the weirdest thing the other day. She said "An apple a day
keeps the doctor away, unquote. Also isn't Eve such a great person?
Like my mom always said quote Eve is the best person ever, much better
than that Alice person." What do you suppose it means?

How would Bob properly quote Eve without causing the above ambiguity?

Note that this problem only really applies to spoken English, since if instead Bob was writing a letter to Alice, he could write

Eve said the weirdest thing the other day. She said "An apple a day
keeps the doctor away, unquote. Also isn't Eve such a great person?
Like my mom always said quote Eve is the best person ever, much better
than that Alice person." What do you suppose it means?

and if Eve instead used quotation marks, there are already conventions for dealing with that.

Punctuation marks themselves are never spoken—only words are spoken.

If Eve said unquote, then you would write that Eve said, "Unquote."

There is no ambiguity. In fact, if you change unquote into a closing quotation mark, then you are misquoting her.

This is further shown by the fact that exactly the same speech can be represented in writing in more than just one way.

Eve said, "Unquote."

'Unquote', said Eve.

Eve: Unquote.

« Unquote dit Eve. [In French.]

There is absolutely no reason why a quotation mark (or a double quotation mark) needs to be used. The same thing can be expressed in different ways. It's just convention that in North America we use double quotation marks. (But quotation marks are not used in plays or movie scripts.)

But what words come out of somebody's mouth do need to be put on the page literally. Doing something different would be an inaccurate transcription of what they actually said.

It's also a convention that (aside from some exceptional circumstances) when writing dialogue, it's words that are used, not numerals or other symbols.

For example:

Mary told John to give her $5.

Mary said to John, "Give me five dollars."

If Mary actually said $5 (assuming she could say a symbol), what would likely have been spoken would be something like this:

Mary said to John, "Give me dollar-sign five."

To resolve ambiguity in spoken English (and, I’d expect, in any other language), use appropriate framing / context.

To use your example, Bob could say:

Eve said the weirdest thing the other day.
She said, “An apple a day keeps the doctor away,” then she said the word, “unquote”. Then she referred to herself in the third person, saying, “Also isn't Eve such a great person?” Eve then quotes her mother by saying, “Like my mom always said quote Eve is the best person ever, much better than that Alice person”.

As you can see, this gets messy pretty quickly.

In practice, it would be easier to quote the relevant parts, simplifying the commentary.