Possessive of "day" for "day's leave" [duplicate]

The fourth example is the correct interpretation of day's, but with two things to keep in mind.

First, in your conclusion you flipped the words around incorrectly*; the journey "belongs to" the day, not the other way around. You could re-write the sentence as:

The house is a journey of a full day from here.

Second, while the journey is "of a day," this does not necessarily mean the day "owns" or "possesses" the journey; grammatically, time periods are simply treated as possessive.

* "A full day of journey" would actually work, but that would make "of journey" a subordinate clause rather than the day belonging to the journey; though, as WS2 brought out in the comments, this should actually be "A full day of journeying".

It's called a "transferred epithet"- the possessive case is incidental, as in...

I had a good night's sleep: The good sleep was mine to enjoy, but it is attributed to the night it happened.

He put in a honest day's work: The quality or extent of work belonged to the doer, but it is attributed to the day.