I would analyze your sentence as follows:
He is the subject.
was is a linking verb.
as big as a house is a subjective complement: "A noun,
noun-equivalent, or adjective placed after a linking verb to complete
the meaning of the subject." ---
Understanding Grammar, by Paul Roberts
The first
as is an adverb modifying the adjective
big.
The second
as is a subordinating conjunction that introduces an elliptic adverb clause:
as a house [is big]. This clause also modifies
big. The "
as...as..." construction works as an adverbial unit.
More explicitly, the meaning is: He was big to the degree that a house is big.