Mike's house is just a stone's throw away from mine.
This means Mike is living next door.
Stone's throw away is an idiom to say something is quite near or at a very short distance. I understand this.
The question is what is an apostrophe doing there? Is it possession? In any case how?
Also, why the indefinite article a is required?
Intuitively, we try to label all uses of the genitive 's as possession, but semantically it just doesn't work. It's true that 's is prototypically used for possession, but it has all sorts of other uses. For example, in an hour's delay, the genitive phrase indicates how long the delay is. Clearly it doesn't indicate that the hour owns a delay. (What would this mean?)
The genitive 's has a great many possible meanings. In fact, it has so many that listing them is famously considered a rather challenging problem:
A classic story in linguistics lore tells of the grammarian who tried to classify all of the ways the genitive can be used. He eventually threw up his hands and said that the genitive is the case that shows any relationship between two substantives.
Indeed, the genitive can be used to express almost any relationship, and even very large lists generally can't be considered exhaustive. Let's take a look at this chart from The Cambridge Grammar of the English Language, p.473-4:
Range of semantic relations
The range of semantic relations between the genitive NP and the head is vast, and largely parallel to that found between subject and predicate in clause structure; some of this variety is illustrated in [the chart below] … NPs are given in the left-hand column, and can be compared to clauses in the middle column, where the semantic relation between the subject and the NP within the predicate is parallel to that between the genitive and the head nominal. The right-hand column contains a generalised schema of each semantic relation:
Mary's green eyes Mary has green eyes. [d has body part h] Mary's younger sister Mary has a younger sister. [d has kin relation h] Mary's husband Mary has a husband. [d has married relation h] Mary's boss Mary has a boss. [d has superior h] Mary's secretary Mary has a secretary. [d has subordinate h] Mary's friend Mary has a friend. [d has equal h] Mary's team Mary belongs to a team. [d is member of h] Mary's debut Mary performs her debut. [d is performer of h] Mary's book Mary writes a book. [d is creator of h] Mary's new house Mary owns a new house. [d is owner of h] Mary's honour Mary is honourable. [d has human property h] Mary's anger Mary feels angry. [d has feeling h] Mary's letter Mary receives a letter. [d is recipient of h] Mary's obituary Mary is the topic of an obituary. [d is human topic of h] Mary's surgery Mary undergoes surgery. [d is undergoer of h] the room's Persian carpet The room contains a Persian carpet. [d is location of h] this year's new fashions This year is a time of new fashions. [d is time of h] the sun's rays The sun emits rays. [d is natural source of h] the cathedral's spire The cathedral has a spire. [d has inherent part h] the war's ancient origins The war has ancient origins. [d has cause h] the flood's consequences The flood has consequences. [d has result h] the lock's key The lock has a key. [d has associated part h] the summer's heat The summer is hot. [d has non-human property h] (d = dependent, h = head)
A stone's throw is a fixed phrase literally referring to the distance a stone can be thrown. It does not indicate possession—a stone is not capable of owning abstract events. Given CGEL's categories, I would categorize the genitive here as expressing [d is undergoer of h]. In other words, I would say it has a patient role:
Patient: undergoes the action and changes its state (e.g., The falling rocks crushed the car.).
This is just one of many possible uses of the English genitive 's.
The indefinite article is required because throw here is a countable noun. Stone's is not acting as a determiner (though genitive noun phrases often do). Compare a [fine summer's] day or an [old people's] home.
It's possessive - the throw belongs to the stone. It's just one of those idiomatic things. The a is used because you're only talking about a single metaphorical stone.
A stone's throw suggests a possessive character, i.e. The throw of a stone.
Regarding the "Old people's home", I've always thought of it as "Old peoples' home"
