First, -120 dBu is 775 nV (7.75 x 10-7 V), not 10-12V. 0 dBu is the voltage (0.775 V) corresponding to 0 dBm (1 mW) in 600 ohms, the traditional reference impedance for audio work.
My original suggestion was to use a primary battery (non-rechargeable) as the voltage reference for a solid-state regulator using a good op amp, stabilizing the voltage from a reasonably clean source. A larger battery (possibly rechargeable) could supply that power at the current required for your load, in theory avoiding any line-related hum. A second circuit could use negative voltages.
There is a large literature about Zener diode noise. The minimum possible noise is the shot noise of the current flowing across the potential drop of the Zener voltage, similar to the shot noise of electrons flowing through a vacuum diode, as opposed to the current flowing through a resistor, which is a different problem. Laboratory noise generators often use a Zener diode as a controllable noise source. There are other "parasitic" noise sources on top of the shot noise.