We show that for a wide class of Gaussian random fields, points are polar in the critical dimension. Examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients, such as the stochastic heat equation or wave equation with space-time white noise, or colored noise in spatial dimensions $k\geq 1$. Our approach builds on a delicate covering argument developed by M. Talagrand for the study of fractional Brownian motion, and uses a harmonizable representation of the solutions of these stochastic pde's.