We observe the elastic stiffness and ultrasonic absorption of a Skyrmion crystal in the chiral-lattice magnet MnSi. The Skyrmion crystal lattice exhibits a stiffness 3 orders of magnitude smaller than that of the atomic lattice of MnSi, being as soft as the flux line lattice in type-II superconductors. The observed anisotropic elastic responses are consistent with the cylindrical shape of the Skyrmion spin texture. Phenomenological analysis reveals that the spin-orbit coupling is responsible for the emergence of anisotropic elasticity in the Skyrmion lattice.