In this paper, we propose for first time practical parameters to construct a compact sub-Angstrom Free Electron Laser (FEL) based on Compton backscattering. Our recipe is based on using picocoulomb electron bunch, enabling very low emittance and ultracold electron beam. We assume the FEL is operating in a quantum regime of Self Amplified Spontaneous Emission (SASE). The fundamental quantum feature is a significantly narrower spectrum of the emitted radiation relative to classical SASE. The quantum regime of the SASE FEL is reached when the momentum spread of the electron beam is smaller than the photon recoil momentum. Following the formulae describing SASE FEL operation, realistic designs for quantum FEL experiments are proposed. We discuss the practical constraints that influence the experimental parameters. Numerical simulations of power spectra and intensities are presented and attractive radiation characteristics such as high flux, narrow linewidth, and short pulse structure are demonstrated.