We prove that there exist compact Ricci-flat Riemannian spaces with isolated conical singularities that are not of orbifold type. In particular, the sectional curvature of these spaces is unbounded from both sides. This is done by determining the metric tangent cones of weak solutions to the complex Monge-Ampère equation on certain singular Calabi-Yau projective varieties, proving that the cross-sections of these cones are smooth and that the weak solution converges polynomially to its tangent cone. Joint work with Song Sun.