Here 3 cylinders intersect following the corners of an octahedra.

 

 

 

With radius 1 the surface of the intersection is

and its volume is a third of the surface.

All not flat corners are identical and measure 1/4. This corners are not different from the corners with three edges in the rhombodoecahedra which can be deformed to our cylinder intersection by pushing the corners with four edges inwards.