In the plane, a Kakeya set is a set of points that contains unit line segments in every direction. Let's define a Kakeya set on an open surface (with a complete metric) to be a set that contains unit length geodesics in every direction. Here, by two geodesics having the same direction, we imply that they stay at finite distance from each other.
Clearly, there are Kakeya sets on any surface: If $\gamma$ is a geodesic so that $d(\gamma(t),x) \rightarrow \infty$, then by taking a sequence of geodesics connecting $x$ to $\gamma(t)$ and deriving a convergent subsequence, we can obtain a geodesic through $x$ in the direction of $\gamma(t)$.
Question: On the plane, there are Kakeya sets of measure zero. Can one construct Kakeya sets of measure zero on any surface as well?