We have experimentally measured transport of superfluid, bosonic atoms in a mesoscopic system: a small channel connecting two large reservoirs. Starting far from equilibrium (superfluid in a single reservoir), we observe first resistive flow transitioning at a critical current into superflow, characterized by long-lived oscillations. We reproduce this full evolution with a simple electronic circuit model. We find that the resistance is consistent with phase slips and the associated creation of vortices, as proposed in [R. P. Feynman, in $backslash$it Prog. Low Temp. Phys., edited by C. J. Gorter (North Holland Publishing Company, Amsterdam, 1955), Chap. 2]. We also show that the oscillations are consistent with LC oscillations as estimated by the kinetic inductance and effective capacitance in our system. Our system allows only a few single-particle, transverse modes to propagate, a situation that, for fermions, would lead to a conductance of only a few $h^-1$. By contrast, in our bosonic system, we observe conductances of thousands of $h^-1$, showing the definitive role played by particle statistics.