  In this paper we investigate the vacuum polarization effects associated with
a massive fermionic field due to the non-trivial topology of the global
monopole spacetime and boundary conditions imposed on this field. Specifically
we investigate the vacuum expectation values of the energy-momentum tensor and
fermionic condensate admitting that the field obeys the MIT bag boundary
condition on two concentric spherical shells. In order to develop this
analysis, we use the generalized Abel-Plana summation, which allows to extract
from the vacuum expectation values the contribution coming from a single sphere
geometry and to present the second sphere induced part in terms of
exponentially convergent integrals. In the limit of strong gravitational field
corresponding to small values of the parameter describing the solid angle
deficit in global monopole geometry, the interference part in the expectation
values are exponentially suppressed. The vacuum forces acting on spheres are
presented as the sum of self-action and interaction terms. Due to the surface
divergences, the first one is divergent and needs additional renormalization,
while the second one is finite for all non-zero distances between the spheres.
By making use of zeta function renormalization technique, the total Casimir
energy is evaluated in the region between two spheres. It is shown that the
interaction part of the vacuum energy is negative and the interaction forces
between the spheres are attractive. Asymptotic expressions are derived in
various limiting cases. As a special case we discuss the fermionic vacuum
densities for two spherical shells on background of the Minkowski spacetime.
