Good question! Is that an assignment question or just something you've wondered yourself?
You'd first have to define what a "straight line" is on this surface. Usually, such a "straight line" is called a geodesic.
Here is a trivial case: if the surface is not path-connected, then it certainly is impossible to go from a point in some path component to some point in another path component.
But if the surface is path connected, I don't know!