High quality Au/hBN/Au tunnel devices are fabricated using transferred atomically thin hexagonal boron nitride as the tunneling barrier. All tunnel junctions show tunneling resistance on the order of several kΩ/μm2. Ohmic I-V curves at small bias with no signs of resonances indicate the sparsity of defects. Tunneling current shot noise is measured in these devices, and the excess shot noise shows consistency with theoretical expectations. These results show that atomically thin hBN is an excellent tunnel barrier, especially for the study of shot noise properties, and this can enable the study of the tunneling density of states and shot noise spectroscopy in more complex systems.