One stage is two diodes plus two capacitors, so that would be a four stage multiplier, with two capacitors in series per stage. This is a much more reasonable number, and very typical for dental X-ray supplies, see the attached image.
The main challenge with operating it in air is that high voltage diodes often don't have enough creepage distance across the package to stand off the rated voltage, so you need something other than air for insulation. If liquids are out, that leaves gases and solids. Gases are difficult to get (SF6) or have other dangers (butane), and sealing them is a pain. Silicone or epoxy can be an option, but they will not remove heat as well as oil so the available power is lower. How much power are you aiming for, and for how long? A dental x-ray tube can typically handle 500 watts for a few seconds, and 250 watts for a few tens of seconds, medical diagnostic tubes can handle more.
Note that 350 - 600 kHz is a pretty high frequency for a multiplier like this, and diode recovery might be problematic, a more typical frequency is in the 50 - 200 kHz range, the multiplier board with the onboard transformer in my picture was made for 150 kHz operation.