He recorded his life there in his diary.

=> He recorded his life in Antarctica in his diary.

1. Which life did he record in his diary?
2. Which life of his did he record in his diary?
- He recorded his life in Antarctica in his diary.
[Which question is suitable for the answer?]

3. Where did he record his life in his diary?
- He recorded his life in his diary / in Antarctica. [3-1]
- He recorded his life in Antarctica in his diary. [3-2]

[If somebody asks this question, can we answer as in the above? In this case, is 'in Antarctica' an adverbial phrase? Can we use [3-2] as a suitable answer?]

In all these sentences, "in Antarctica" is an adjective modifying "life."

Since we all have only one life, I wouldn't use either 1 or 2. I'd ask, "What part of his life did he record in his diary?"

If you want "in Antarctica" to be used as an adverb, then here's how:
~ In Antarctica, he recorded his life in his diary.

Remember, as I've written before -- the adjective or adverb needs to be as close to what it's modifying as possible to be crystal clear.

Yes, you can answer the question "Where did he record his life in his diary?" by saying "He recorded his life in Antarctica in his diary." In this sentence, "in Antarctica" is indeed an adverbial phrase that specifies the location where he recorded his life. Both [3-1] and [3-2] can be suitable answers to the question, depending on the specific context or emphasis.