We investigate the question of approximability of Gaussian states by Gaussian tensor networks. For a simple 1d free fermion model, we prove that any Gaussian MPS approximation to its ground state must have bond dimension growing superpolynomially in system size, while it is known that a non-Gaussian MPS approximation with polynomial bond dimension exists. This shows that, in general, imposing Gaussianity at the level of the tensor network may significantly alter its capability to efficiently approximate critical Gaussian states. We also provide numerical evidence that the required bond dimension is still subexponential. Based on joint work with Ignacio Cirac (arXiv:2204.02478).