Antipodal sets in a Riemannian symmetric space is defined in a geometric way by the use of geodesic symmetries. The oriented real Grassmann manifold is a Reimannian symmetric space and antipodal sets in it correspond to certain combinatorial objects. In the case where the rank of the oriented real Grassmann manifold is less than five we give the classification of antipodal sets in it. In the case where the rank is equal to five we determine antipodal sets of maximal cardinality. We mention a recent result of Frankl and Tokushige in the case of higher rank.