Five-dimensional space of non-rectilinear Keplerian orbits is considered, as well as four its quotient spaces. In the last ones orbits are identified irrespective of values of longitudes of nodes, values of arguments of pericentres, values of both longitudes of nodes and arguments of pericentres, values of longitudes of nodes and arguments of pericentres under fixed values of longitudes of pericentres. All these spaces (except the last one) becomes metric spaces by introducing a suitable metric. Usable formulae for calculation of distances between orbits via their Keplerian elements are given. As to the last quotient space, the constructed for it function of a pair of orbits satisfies first two axioms of metric spaces. The validity of the third axiom (triangle axiom) is not demonstrated or disproved yet. The introduced orbital spaces, together with metrics, serve as a good tool for problems of searching close orbits, and identification of parent bodies in comet-asteroid-meteoroid complexes.