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".
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'.