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.

Update: Alejandro, thank you for your comment. I don't think though, that connectedness (or path-connectedness) would make things better. Again, for any connected (path-connected) Y and any line-filling space X, f: X \to [0,1] take x \in X and y \in Y and glue X and Y at the point (x,y) (that is consider equivalence relation with only one non-trivial equivalence x \equiv y and take the factor space). Denote the result by Z. Then one extends f to Z by defining f(Y) = f(x). I believe Z is then connected (if X and Y were) and again pretty random.