The Hilbert curve is a classic example of a family of curves whose limit fills the unit square, which can be then extended to the whole plane. There are analogues of the curve in three dimensions, and one could supply a space filling curve in any finite number of dimensions, it seems, by simply extending the Hilbert curve.
Could there be a space filling curve in an infinite dimensional space? Or are there too many degrees of freedom for just a line to fill it all? Countable, uncountably infinite spaces?
For uncountably many dimensions:
Consider the vector space $V$ of functions from $\Bbb R$ to $\Bbb R$. Its cardinality is greater than that of $\Bbb R$ (by a diagonalization argument, I think). So is the cardinality of the "unit square" in that space. Hence there cannot be a surjective map from the real line onto that unit square.
So in at least one case, the answer is "no."