Is there anything wrong with this (details inside)

Your sentence makes sense with the two thats, but I would have written “that it” was impossible instead. I use “that” too much in my writing to begin with, so whenever I can delete a that, or substitute a different word, I do it.

At times it is simple to use another appropriate word. As @JLeslie said you can replace the second “that” with “it”. Occasionally an immediate repetition of a word, separated from the first instance by punctuation, is appropriate for a better effect. “I tried to persuade him, but he said that, that was impossible”.

There is nothing wrong with the sentence…

Same words can happen. In Dutch we have a sentence that goes like this:

“als achter vliegen vliegen vliegen, vliegen vliegen vliegen achterna”.

In English there is buffalo that buffalo Buffalo buffalo, if they buffalo other Buffalo buffalo.

No. do this ‘I told him that wasn’t possible.’

One of the more famous poems in English, The Rime of the Ancient Mariner, contains the lines:
“Water, water, everywhere,
And all the boards did shrink;
Water, water, everywhere,
Nor any drop to drink.”

From a stylistic point of view it is generally inadvisable to repeat the same word in quick succession. Even the adverb “very” shouldn’t be used in succession: “How fat was he? He was very, very fat.” (There it just indicates the writer’s laziness or lack of imagination: He was monstrously, grotesquely obese, not “very very fat”.)

So I’d go with what @JLeslie said and recast the sentence for stylistic reasons, although there is nothing grammatically ‘wrong’ with your original sentence despite your friend’s strong condemnation. I am so, so sick of people who don’t really understand the rules pretending that they do and enforcing them as if they did.

PS: Unfortunately, @Smitha‘s suggested “correction” of adding a comma between the words would actually have been an error.

It is fine…just use sparingly (amd never separate them with a comma).

Porky pig said it loads of times, “Th…thth…thhthth…that’s all folks!”

@CWOTUS Oh, I thought we could use it like that too. Thanks for pointing out the error. I think I’ll flag it.

@Smitha: Your use of written English is almost perfect and much much better than that of many native speakers. That that’s true I can attest.

I can’t believe that that woman would tell you that, that being incorrect.

@gailcalled Thanks for that! It could be because English is my secondary language and I keep checking everything twice. I always tend to avoid grammar questions.

@ibstubro I think grammatically it’s incorrect, but I thought we could use it to avoid confusion. No worries, I have flagged it as not helpful, but it’s still there.

No, I was not directed @ you, @Smitha. I was just pointing out that that kind of usage is okay.

What @gailcalled said.

Regarding what @gailcalled in complimenting @Smitha on the quality of her written English, I will also attest that that is so. So there.

And that’s that.

@Smitha:Your prose is not just correct but stylish and economical, too. Some flutherers are guilty of tediously and tiresomely lengthy answers. I am going to add “prolix” to the word-of-the-day question.

Like @JLeslie I am also inclined to use the word “that” too often, these days I read through what I have written and delete any superfluous “that’s”, more often than not a sentence makes perfect sense without the “that”.

@Adagio & @JLeslie I’ve been doing (or trying to do) the exact same thing. Once you become conscious of doing it, it’s really annoying.

Two ‘that’s’ together may not be wrong but sound cumbersome. Why not just eliminate one….”.I tried to persuade him, but he said that was impossible”.