An animal embryo begins its life as a ball of epithelial cells. In the course of development, invariably, this cellular ball will undergo a process of gastrulation to form a multilayered structure with the different germ layers designated to form organs with different shapes and functions. In the fruit fly Drosophia melanogaster, gastrulation begins with the constriction of mesodermal cells that make up a rectangular domain in the ventral part of the embryo. A remarkable aspect of this morphogenetic event is its anisotropy - the mesoderm constricts much more along one axis than along the other. In this paper we propose an explanation of this observed anisotropy. Specifically, we show that tissue contraction must be anisotropic, provided that the tissue is elastic and that the contractile domain is elongated (e.g. rectangular as opposed to square). This conclusion is generic in the sense that it does not depend on the specific values of model parameters. Since our recent study demonstrated that embryonic tissue is elastic on a developmentally relevant time-scale, it appears likely that the anisotropy of mesoderm contraction is an elastic effect. Our model makes a number of specific predictions that appear in close agreement with the available data.