It was one of a few houses (that were) (fully) built.
I want to say that the house was done, does just built mean that or should it be fully built ?
and is that were necessary or optional?
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The identifying relative clause of the example sentence is highlighted below:
It was one of a few houses that were fully built.
According to the Oxford Guide to English Grammar, Eastwood (1994) p. 361:
Sometimes we can use a participle without a relative pronoun or an auxiliary.
That man sitting next to Angela never said a word.
In your example sentence, you can use a passive participle (built) in a reduced relative clause. Note that this usage is common in informal language.
These are fine:
It was one of the few houses fully built.
It was one of the few houses built. (modifier can be removed)