  We study configurations consisting of a gravitating spinor field $\psi$ with
a nonlinearity of the type $\lambda\left(\bar\psi\psi\right)^2$. To ensure
spherical symmetry of the configurations, we use two spin-$\frac{1}{2}$ fields
forming a spin singlet. For such systems, we find regular stationary
asymptotically flat solutions describing compact objects. For negative values
of the coupling constant $\lambda$, it is shown that, by choosing physically
reasonable values of this constant, it is possible to obtain configurations
with masses comparable to the Chandrasekhar mass. It enables us to speak of an
astrophysical interpretation of the obtained systems, regarding them as Dirac
stars.
