When a vehicle or a vessel moves through air or water at a steady speed, most of its fuel is spent on fighting the hydrodynamic resistance. This is because the vessel needs to push the fluid out of its way in order to move forward. The disturbance of the fluid caused by the vessel motion normally spans the distances greatly exceeding the size of the vessel; enormous amounts of energy are needed to displace the huge mass of fluid in that range. Using computer fluid dynamics (CFD) simulations, we show that the volume where the fluid is pushed by a vessel can be reduced to a thin layer filled with a properly designed fluid-permeable porous metamaterial with anisotropic permeability. The structure, dubbed “fluid flow cloak,” moves through the fluid without generating any wake behind it, and it experiences zero resistance force, known as the drag force in hydrodynamics. The drag force normalized to the size of the vehicle, or the drag coefficient, is one of the most important measures of fuel efficiency. For relatively slow motions, we find that the drag coefficient can be made to vanish.