神様のいない日曜日 is one of my favorite animes!
Well, 神様のいない日曜日 is the same in meaning as 神様がいない日曜日.
In relative clauses, we can use の instead of が as a subject marker. (If you want to know why, this question may be of help: Why can の and が both mark subjects in relative clauses?)
Here, 神様のいない is a relative clause that modifies 日曜日.
So this の is not of possession, and does not have the nuance of "X owns Y."
Since the meaning of 神様がいない is "God does not exist", the meaning of 神様のいない日曜日 (== 神様がいない日曜日) is "Sunday on which God does not exist".
By the way, your attempt "神様いないの日曜日" makes no sense.
First, 神様いない is not grammatical. As 神様 is a subject here, it should be 神様がいない.
When you want to add information of the form of a complete sentence (like 神様がいない) to a noun, you can't use の. To do that, make it a relative clause. By doing so you get the phrase 神様がいない日曜日.