The flow problem of an incompressible axisymmetrical quasisteady translation and steady rotation of a porous spheroid in a concentric spheroidal container are studied analytically. The same small departure from a sphere is considered for each spheroidal surface. In the limit of small Reynolds number, the Brinkman equation for the flow inside the porous region and the Stokes equation for the outside region in their stream functions formulations and velocity components, which are proportional to the translational and angular velocities, respectively, are used. Explicit expressions are obtained for both inside and outside flow fields to the first order in a small parameter characterizing the deformation of the spheroidal surface from the spherical shape. The hydrodynamic drag force and couple exerted on the porous spheroid are obtained for the special cases of prolate and oblate spheroids in closed forms. The dependence of the normalized wall-corrected translational and rotational mobilities on permeability for a porous spheroid in an unbounded medium and for a solid spheroid in a cell on the particle volume fraction is discussed numerically and graphically for various values of the deformation parameter. In the limiting cases, the analytical solutions describing the drag force and torque or mobilities for a porous spheroid in the spheroidal vessel reduce to those for a solid sphere and for a porous sphere in a spherical cell.