Aleksandr Solzhenitsyn wrote the novel: Оди́н день Ива́на Дени́совича.
It has always, to my knowledge, been translated as:
A) One Day in the Life of Ivan Denisovich
which, as others have already answered, is perfectly natural and idiomatic in English. A day in the life of is indeed quite common, so common it can be considered a fixed phrase.
Else, you would have:
B) One day in Ivan Denisovich's life, or even, word-for-word, One day of Ivan Denisovich's.
which do not have the same ring (rhythm) or zing (energy) to it. The idiomatic title (A) has a rhythm to it and English has a certain rhythm, or series of stress points, built into it. One can clap one's hands to many English sentences. And if the choice is between a title or construction that is rhythmic and one that is lesser so, a wise publisher will go with the former.
Else, you also have:
C) A Day in a Farmer's Life.
Which sounds just so bland in comparison to the 'normal' way of formatting the phrase. It is so uninteresting a title (and phrase) that I might not even look at the book if I had the choice of others to look at.
In other contexts, the pattern could stress whose thing, for example, whose answer it is:
If I say, (D) rogermue's answer, I am using the name as an adjective and thus am not really calling attention to the fact that it is (E) the answer of rogermue, which somewhat calls attention to the fact that rogermue (and not someone esle) wrote the answer.
Last, notice the definite article in the preferred phrases (B) and (C) and the specialized phrase (E): the adds a certain punch in the phrase. We know we are talking about a specific life or a specific answer. It is hard to explain what the little the does there, except to say that perhaps we all prefer to read about the life or read the answer rather than about a life or so-and-so's life or so-and-so's answer.