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Let's say someone has asked a question: "How can one do X?"
But it turns out in fact that one cannot actually do X. So someone posts a response saying that X is impossible to do.
Can this response be considered an answer to the original question or be considered to be answering the original question?
For example, would it be correct for the person who posted the question to say: "Thank you, my question has been answered"?
It depends upon your definition of 'answer'.
According to Oxford Languages, 'answer' can mean:
A: "A thing that is said, written, or done as a reaction to a question, statement, or situation".
or
B: "A solution to a problem or dilemma".
Definition A describes a response, whereas Definition B describes a response containing a solution.
If someone was to ask the question, "How can drive a car if I am dead?", and another person responded by saying, "Driving a car when you're dead is impossible", they have satisfied definition A, but not definition B.
If someone was to respond by saying, "Driving a car when you're dead is impossible, but you could do it if you could do the impossible, by turning yourself into a zombie", they have provided a hypothetical solution, thereby possibly satisfying definitions A and B.
Note however that there is an important discrepancy between the phrasing of your title question and the material beneath, for to state that the 'question is unanswerable' is different to stating that a thing is 'impossible to do'.
A question that is literally unanswerable simply cannot be answered, regardless of the form the answer entails.
However, we know that (barring some kind of incapacity), a person can always react in some way to a question, so the statement that a question is unanswerable seems only to apply to situations in which no-one was present to react, or in situations where no-one was capable of reacting. But of course, this would mean that whoever poses such a question must somehow be rendered incapacitated or non-existent after uttering it, for they would otherwise be capable of providing an answer.
The reply, "This question is unanswerable" cannot exist in response to a truly unanswerable question, for such a reply requires an answerable question in order to made.
"This question is unanswerable", can be made in response to any question other an unanswerable question. It will however be an inaccurate response, for it will have fulfilled at least one definition - definition A - of the word 'answer'.
The question, "How can one do X?", can be asked in two contexts.
1. Knowing that it is unanswerable
2. Without knowing whether it is unanswerable or not
In the first case your response is sensible.
But in the second case your response is not sensible because it is not the answer to the original question. Even when the answer was given with proof, even that answer doesn't answer the original question.
When roughly speaking, the second case is of course a response to a question. But when strictly speaking...,since the OP was looking for a specific way, he didn't get the specific way he was looking for. So we can say that he got an answer to the question, but didn't get an answer to the original question. Similarly we can say that he didn't get a solution to his original problem. But don't forget that in the first case this is the answer.
Even though wrong answer as well as right answer are answers, we often do not give compliment for the wrong answer. Instead, we might thank for the response; not for the answer. Similarly, for an unanswerable question, we might thank for the response or effort; not for the answer. I mean, in these cases, the thanks we give is actually not for the answer; but only for the time spent for us or for the effort. But if we got the answer to the original question, a smidgen would be for the answer.
If you are willing to think about the second context, you will get two cases.
2.1 The questioner doesn't know whether it is unanswerable, but will be satisfied with any final answer.
2.2 The questioner doesn't know whether it is unanswerable, but will not be satisfied with all final answers. What he needs is a solution to his particular problem.
(Though I don't like) You may treat 2.1 as you treated context 1 (That is, as 'sensible')
In a logical sense the question would not be answered, instead you made an observation about the question itself. An observation is not an answer.
In fact the observation is the answer to a question that was not asked: "does //the question in reference// have an answer?"
