I'll try to offer a little less technical explanation.
Don't think of these の particles as being possession. Think of them as one noun modifying another. We take the first and mix its flavor with the second one, if you will. So for example:
It seems a little fuzzy but this is essentially packing a whole bunch of descriptions of 家 into one sentence. It's awkwardly long and that's kind of the point, just like the English translation being a long list of characteristics is also long and unwieldy. Let's take another approach from the inside out looking at how these の are working.
At the core, we have 住みやすい家. No particle trickery here. A house that's 'easy to live in.' Adding on to that we have [窓の大きい]住みやすい家. This is referring to a livable house with big windows. If we mix the meanings together the meaning is clear. In this particular situation the の functions like the particle が. This is a slightly older pattern that still shows up in some places, like 髪の長い女の子, or a girl with long hair.
Moving out from the original nucleus of 住みやすい家 and adding on more and more of the ～の phrases adds on more of that style of modification. So next you'd add 三階建ての. This is what Darius is talking about with the である stuff. You would normally want to say 三階建てです, but you can't use だ/です to connect to another word. So we use である to allow the copula to connect to other things, and the の in this place has the same role as that である.
So from there we can continue adding extra characteristics. Lump all the previous ones together and let's just call it 家. So now we have just added the idea of 3 floors to it. Then to our new 家 language blob we add 庭付き to talk about its yard/garden in a fashion similar to the 三階建て.
Note that the 静かなところにある uses に and not の. This is because の does not refer to a location. If we were to say 静かな所のある then we would say that the house (or possibly the 庭, it's a little ambiguous) has a quiet place within it, as in the の/が pattern I mentioned above.
I would also suggest that 名古屋市内の静かな所 should be interpreted separately as its own unit. This livable house with big windows and three floors and a garden is in a quiet place in Nagoya. Perhaps you can call this a genitive の, or what you have been associating it with as the possessive の.
This is a very basic explanation and I am not striving for technical precision down to the most minute detail. Nevertheless, if you think of の as a general idea of noun modification rather than just possession it will start to make more sense.