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That depends - are you using it as an electrical insulating oil? In that case, most manufacturers specify a maximum around 30 ppm.

For lube blending, the maximum allowable amount of water in any base oil is determined by your process - how much can you tolerate before the end product is hazy? You'll only find this out experimentally, by making bench blends to see what level of water causes you a problem, which then sets your spec. Then you find a supplier with product that is lower than that level.
Because of their superior solvency, a naphthenic will hold a more water without exhibiting a haze than a paraffinic.

35 ppm is the industry spec for transformer oils. They need to be very dry for obvious reasons. Due to the very critical spec, test precision-related variation between labs and the fact that water just seems to find its way into the most well-protected trailers, most reliable suppliers won't release naphthenic oil intended for transformer service if the water content measures over 30 ppm.
Cleanliness specs as in particle count? I don't think there are any. Particle count could be deceptive, as not all particles would be similarly conductive. As long as the oil passes D877 (Dielectric Strength) and D924 (Dielectric Constant/Power Factor), particulate levels seem sort of irrelevant. If the oil insulates, that's really all that matters. The contaminants of concern are water, corrosive sulfur and acidic components - all things that decrease insulating value.

Since these oils are typically in non-circulating applications (except maybe for very large substation Type I applications) any particles are going to wind up at the bottom of the can after a little while.

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