The current-voltage and noise characteristics of vacuum nanoelectronic devices, including nanoscale field-emission diodes and vacuum nanotriodes, are presented. The turn-on voltage for field emission in these devices is about 8 V, independent of ambient temperature, and currents of up to 10 nA are obtained. Time-independent fluctuations are observed in the current-voltage characteristics of the vacuum nanoelectronic devices that persist down to 20 K. In the case of the nanotriode, these fluctuations are repeatable in different sweeps performed both at the same and at different temperatures. These measurements, coupled with a model of the nanotriode, which includes an analysis of the electrostatic field within the nanochamber, the electron trajectories, and the role of field-emission resonances, suggest that the origin of the time-independent fluctuations is a quantum-interference effect between the electron wave function in the tip of the nanopillar field emitter and the anode.