Using "about whether" in a sentence

Yes, it is correct. You can also say "on" instead of "about". Here you'll find some examples from published books. https://books.google.com/ngrams/graph?year_start=1800&year_end=2019&smoothing=3&content=a+debate+about+whether%2Ca+debate+on+whether%2C&corpus=26&direct_url=t1%3B%2Ca%20debate%20about%20whether%3B%2Cc0%3B.t1%3B%2Ca%20debate%20on%20whether%3B%2Cc0

It is quite common to have a bad situation, however, nowadays there is an intense debate about [whether it is better to keep trying until obtaining what you want, or just to accept the things as they are].

Yes, it's correct.

"Whether" is an interrogative (question) word introducing the bracketed subordinate interrogative clause (embedded question) functioning as complement of "about".

The meaning can be glossed as:

"... nowadays there is an intense debate about the answer to the question 'Is it better to keep trying until obtaining what you want, or just to accept the things as they are?'"

You're focussing on the wrong thing. "About whether" is not a constituent; it's not a phrase, or a clause, or a construction. It's not anything by itself and for sure there's no rule about it. It's just two words that happen to occur together in a sentence. Both of them introduce constituents, and those constituents are stacked inside one another like Russian matryoshka.

The operative clause (the rest of the sentence is not involved) is

• there is an intense debate [about [whether [it is better to keep trying]]]

The brackets show the embedded constructions modifying debate:

1. the tensed clause: it is better to keep trying (there's more clause in the original)
2. the embedded question: whether it is better to keep trying, introduced by whether
3. the prepositional phrase: about whether it is better to keep trying, introduced by about

Whether introduces an embedded Yes/No question

(whether is the Wh-word for Yes/No questions; it's deleted with a real question,
but retained when they're embedded).

This particular Yes/No question is

• Is it better to keep trying?

but of course an embedded question doesn't do subject-auxiliary inversion -- just the Wh-word is enough.

About is a preposition, and therefore introduces a prepositional phrase -- its object has to be a noun phrase, and embedded questions are noun phrases. Embedded questions are one of the four types of complement ("noun") clauses that function as nouns -- subject, object, prepositional object.

So in this case the preposition about takes as object a constituent clause that starts with the word whether, and that word in turn introduces another constituent clause. It's a complex situation, all right, but focussing on two words that don't belong together isn't going to help.

Look for constructions and clauses, not individual words; English sentences are not stuck together like beads on a string. They're constructed more like large buildings, with parts that fit together and support one another.