What is typically being made reference with that sentiment are one-to-one interpretations. One word in Language A for one word in Language B. That much is certainly true, but not the idea that certain words resist all attempts at translation. If Language A and Language B are both true, organic human languages, then that will never be the case.
Here's one that I, and a number of other translators struggled to do, again from Japanese. The written or literary form of a sentence (in a story, for example) is different from the spoken form. So, "the house is red", when spoken, is: "ie wa akai desu" Written in a story, it would be "ie wa akai de arimasu" Which is fine, because normally, you just ignore the sentence ending particle and translate both as "the house is red". But in one particular anime, there was a character who used the literary form when speaking. In other words, she was speaking like a narrator, but also using the form when speaking directly to another character. Several translators just gave up and ignored it. Some translators included the sentence endings in the subtitles, which made them somewhat hilarious. I ended up translating them as if she was narrating a third person close POV story - which wasn't entirely accurate, but I felt it was better than the other two options. Sometimes, you just have to make compromises.
There is a similar dynamic in Spanish. You may well know that Spanish has formal and informal, the famous T-V Distinction of Indo-Eupopean languages. Less well known is the fact that this system of formality is much more layered than just tú vs usted. In some regions, tú sits side by side with vos, where, when both are present and vos is not simply replacing tú, a three-tiered system of formality is invoked with vos being the least formal, tú a bit more formal, and usted being formal. But that's not all, folks! What you may never have seen or heard is the truly involved system of formality where one purposefully debases oneself in ever-more complex and baroque 3rd person. One is never "I", but instead (in Spanish), "This humble servant graciously permits himself the honor of sending to the most high and regal Embassy of the United Kingdom the following folios..." And that's just the intro! (Looking at you, judiciary system of the Republic of Colombia and your seeming fetish with keeping the language of Cervantes alive.) Things like that do prove challenging, mostly in the vagaries of what the client wants to see. Some want you to clean all that up because that kind of language comes off as sarcastically servile in English. We certainly have ways of saying those things (I just did it) but we simply do not speak that way and the intended flood of formal honor turns into the polar opposite. But again, there are ways around all of this that answer to situation and circumstance.
Rewriting all my stories in English is a great challenge for me. English is not my mother tongue, and for me a great opportunity to improve myself.