One of the functions of "の" is "~'s(possession)," but that's not the all. You can break down せかいの中の日本 into two parts(1 noun phrase and 1 noun) connected by の: 「せかいの中」の「日本」. This の(AのB) means "B located at A." Again, せかいの中 is broken down into two nouns: 「せかい」の「なか」. This の（CのD）means "D of C". So the whole phrase becomes "Japan, located inside of the world". To make more sense in English, I'd translate it like "Japan amongst the international society" .

せかいの中 = Connecting the two nouns (noun + の + noun) The second の means "of" So the sentence actually means "Inside the world of Japan".

It really means just "Japan". And yes, particle "no" means "...'s".