I feel like the stereotype about this question is different from your approach, though: supposedly, it started with quirky, new tech-minded businesses using it rationally to see people who could solve open-ended problems, and evolved to everyone using it because it was the popular thing. If someone still uses it today, I would totally expect the interviewer to have a number up on their screen, and answers that are too far off would lead to a rejection.

Besides, it's too vague of a question. If I were asked it, I would ask so many clarifying questions that I would not ever be considered for the position. Does "fill" mean just the human/passenger spaces, or all voids in the plane? (Cargo holds, equipment bays, fuel and other tanks, etc). Do I have access to any external documentation about the plane, or can I only derive the answer using experimentation? Can my proposed method give a number that's close to the real answer (if someone were to go and physically fill the plane), or does it have to be exactly spot on with no compromises?