A modern variation of the Rutherford experiment to probe the tunneling of exotic nuclear matter from the measurement of the residues formed in the bombardment of (197)Au by extremely neutron-rich (8)He nuclei is presented. Using a novel off-beam technique the most precise and accurate measurements of fusion and neutron transfer involving reaccelerated unstable beams are reported. The results show unusual behavior of the tunneling of (8)He compared to that for lighter helium isotopes, highlighting the role of the intrinsic structure of composite many-body quantum systems and pairing correlations.