We construct an exactly solvable quantum impurity model which consists of spin-1/2 conduction fermions and a spin-1/2 magnetic moment. The ground state is a Gutzwiller projected Fermi sea with nonorthonormal modes and its wave function in the site-occupation basis is a Jastrow-type homogeneous polynomial. The parent Hamiltonian has all-to-all inverse-square hopping terms between the conduction fermions and inverse-square spin-exchange terms between the conduction fermions and the magnetic moment. The low-lying energy levels, spin-spin correlation function, and von Neumann entanglement entropy of our model demonstrate that it exhibits the essential aspects of spin-1/2 Kondo physics. The machinery developed in this work can generate many other exactly solvable quantum impurity models.