There is, probably, no hope for a similar characterization since for any topological space Y and any X, which is line-filling in your terminology, if f: X \to [0,1] is continuous and surjective then h: X \cap Y \to [0,1], where X \cap Y is a sum of spaces (=disjoint union) X and Y, given by h|_X = f and h(Y) = 0 is also surjective and continuous. So your spaces maybe as bizzare as you want.