There is, probably, no hope for a similar characterization since for any topological space Y$Y$ and any X$X$, which is line-filling in your terminology, if f: X \to [0,1]$f: X \to [0,1]$ is continuous and surjective then h: X \cap Y \to [0,1]$h: X \sqcup Y \to [0,1]$, where X \cap Y$X \sqcup Y$ is a sum of spaces (=disjoint union) X$X$ and Y$Y$, given by h|_X = f$h|_X = f$ and h(Y) = 0$h(Y) = 0$ is also surjective and continuous. So your spaces maybe as bizzarebizarre 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$X$,1] $f: X \to [0,1]$ take x \in X$x \in X$ and y \in Y$y \in Y$ and glue X$X$ and Y$Y$ at the point (x,y)$(x,y)$ (that is consider equivalence relation with only one non-trivial equivalence x \equiv y$x \equiv y$ and take the factor space). Denote the result by Z$Z$. Then one extends f$f$ to Z$Z$ by defining f(Y) = f(x)$f(Y) = f(x)$. I believe Z$Z$ is then connected (if X$X$ and Y$Y$ were) and again pretty random.
