Does the Ricci tensor actually change volume of a body in reality?

Gravitational wave detectors work by measuring tiny differences in the length of the path of laser beams caused by ripples in space-time. So yes, curvature can change the length of a path and the volume of a region.

For example, imagine a box in free-fall above the surface of the earth. For simplicity, imagine it is going along a path straight toward the earth's center (falling straight down). If each corner of the box travels along its geodesic (its natural path through space-time), the corners of the box will be drawn closer together as they each travel toward the center of the earth. Now, if there are electromagnetic forces forcing the box to hold its shape, the box may not change in volume. It will, however, feel the force of compression, and that force will have to be balanced out by some other force or the box will contract. If you just have a free falling laser and mirror setup, the time it takes to the laser light to bounce off the mirror and arrive back at the laser will be shorter in the gravitational field than in flat space.