  We consider the Casimir interaction between two spheres corresponding to
massless Dirac fields with MIT-bag boundary conditions. Using operator
approach, we derive the TGTG-formula for the Casimir interaction energy between
the two spheres. A byproduct is the explicit formula for the translation matrix
that relates the fermionic spherical waves in different coordinate systems. In
the large separation limit, it is found that the order of the Casimir
interaction energy is $L^{-5}$, where $L$ is the separation between the centers
of the spheres. This order is intermediate between that of two Dirichlet
spheres (of order $L^{-3}$) and two Neumann spheres (of order $L^{-7}$). In the
small separation limit, we derive analytically the asymptotic expansion of the
Casimir interaction energy up to the next-to-leading order term. The leading
term agrees with the proximity force approximation. The result for the
next-to-leading order term is compared to the corresponding results for scalar
fields and electromagnetic fields.
