If you look down from space on a man walking along one of these geodesics towards the boundary, he would keep shrinking and his steps would look increasingly smaller. For him it would take infinitely many steps to get to the boundary. This is true even if he walks at what he considers to be constant speed. So for him the geodesic is infinitely long.
the same geometry comes from a plane geometry in which the parallel postulate is false.
The geodesics are just straight lines in this axiomatic version and like any line in a plane geometry they are infinitely long.