Subordination in Chinese Linguists claim that subordination is universal across the world languages. 

Subordination in English looks can be understood by looking into these examples:

I know a person who has a dog

I know a person who has a dog which barks at a cat

I know a person who has a dog which barks at a cat which lives in a house

I know a person who has a dog which barks at a cat which lives in a house which is located in a city which ....

You get the idea...

I would like to know how could we translate the longest English sentence from the examples into Chinese (with preserving the subordination), that is, leave everything in one sentence.
like:

我知道一个的人。

我知道一个有狗的人。

I am no Chinese nor English native speaker, but my intuition is telling me that this sentence, however unnatural, sounds more naturally in English than in Chinese, is that the case?

I even have doubts if subordination really exists in Chinese...
If I extrapolate what you are doing, it actually becomes "unnatural" to me, because the subordinate clause is inserted in front of the noun that is being modified, but after the original verb. Your examples can be translated like the following:

I know a person.
我認識某個人

I know a person who has a dog.
我認識某位有養狗的人

I know a person who has a dog which barks at a cat.
我認識某位養有會對貓吠的狗的人

I know a person who has a dog which barks at a cat which lives in a house.
我認識某位養有會對那隻住在屋裡的貓吠的狗的人

I briefly add/tweak some charaters so that it makes more sense to me.... I'll leave it up to you to see if you can just "get the idea". :)

If I "naturally" say these, it'll probably not be all subornate clause but shorter SVO phrases:

I know a person
我認識某個人

I know a person who has a dog
我認識某個人，有養狗

I know a person who has a dog which barks at a cat
我認識某個人，養的狗會對貓吠

I know a person who has a dog which barks at a cat which lives in a house
我認識某個人，那人養的狗會對貓吠，那貓還住在房子裡