Writers use “untranslatable” as a device to evoke a vague mysticism surrounding a language. IMHO it lends a flavor of superiority over the reader which I find gauche.
You can explain the meaning of a text, sure, and for technical texts that may be all you need, but if the goal is not simply the transfer of dry information and the result fails to trigger the intended associations and emotions, your job as a translator is not yet done.
Personally I prefer footnotes in those situations. I'd rather have a few sentences that explain the nuance an author was trying to convey with their word choice that a native speaker would have picked up on rather than have a phrase thrown out entirely and replaced with some alternative localization that might be familiar in an effort to try to capture the same "feel" but barely comes close to what was actually said (if you're lucky).
To declare something "untranslatable" is essentially to imply that a non-native speaker is incapable of understanding it no matter how much they study the language. It deals in "touchy feely unquantifiable nonsense".
Noticed that everyone in here who proclaims that some things are untranslatable have yet to provide actual concrete examples of them (SRC LANG, TGT LANG, SRC SENTENCE)
The fact that you can have legal documents translated from one language to another is all the proof you need that if you're a good enough translator, all languages are interchangeable. You might need a whole lot more words or a whole lot less words depending on the source and target language though.
Here's an easy one: nearly any pun is literally impossible to translate while still being a pun. You can explain the pun by talking about the original language, but it can't be adequately translated.
I get your irritation but I think it's also a way of signaling to the reader that the difficult word is heavily freighted with meaning that would be blindingly obvious to a native/fluent speaker but require a small essay to convey in English (and still wouldn't have the emotional impact) - linguistic operator overloading, if you will.