The tight span of a finite metric space is essentially the ‘smallest’ path geodesic space into which the metric space embeds isometrically. In this situation, the tight span is also contractible and has a natural cell structure, so that it lends itself naturally to the study of the Cayley graph of a group. As a first step in this study, we consider the tight span of a metric space which arises from the graph metric of an antipodal graph. In particular, we develop techniques for the study of the tight span of a graph, which we then apply to antipodal graphs. In this way, we are able to find the polytopal structure of the tight span for special examples of antipodal graphs. Finally, we present computer generated examples of tight spans which were made possible by the techniques developed in this paper.