"Our family is big."
My guess is that うちの is being used for emphasis. I could probably just say "家族は木きいです = Our family is big." Even though it's not explicitly stated, the assumption is that I am talking about my own family.
On the other hand, "うちの家族は木きいです = Our family is big." In this case, I'm stating explicitly that I'm talking about my family.
Does that answer your question? Or were you just wondering about the placement of うちの?
As has been explained, "uchi" is the kind of "our" that we would use to describe "our house" or "our refrigerator." In English, we just use "our" but this has probably been shortened from "our household (object)" or "our family (object.)" So by "uchi" you're talking about the kind of "our" where the whole family/household shares the thing you're talking about. Other forms of "our" (such as "my friends and my school") would just be watashitachino, which is "the thing that belongs to me and the company with/associated with me" as opposed to "the thing that belongs to the people I live with and me."
In short, not all languages have a 1:1 translation. Often times, we are translating intent and meaning rather than words, since words are meant to convey these ideas in the first place, and languages evolve very differently from one another, especially when having developed independently of one another.
That said, I am still a learner. If I got the watashitachino/uchino bit wrong somehow, someone please correct me.
you should read about the concept of 内・外 from japanese:
Let's start to faze out hiragana in the answer selections for content words (ie high frequency nouns, verbs, adjs) or at least offer Kanji as an option - just a suggestion for administrators because one goal I think many have for Japanese is to transition away from Hiragana and learn to read the language as it shows up in signs and easier books. I think at this level we are ready..
Unless I'm mistaken, "うち" means "house/home", so i've been reading these types of sentence; "うちの..." as "the home's..." with possession implied...
But "the home's family is big" doesn't feel quite right, not like "うちの本棚大きいです / the home's bookcase is big" does...
Anyone able to clarify for me?