No thickness of gold that Hazou can wear and still move around will provide meaningful protection from the amount of X-rays we're talking about here.
Adding onto this: using this and this, for 1 nm X-rays (~1.2 keV), I think that a tenth of a millimetre (100 μm) of gold would be enough to block ~all of the output from a 1nm runic HOWS (comes out for me as intensity reduced by a factor of about 10^84). This seems intuitively very weird, but remember that gold leaf has a thickness of <1 μm and still looks largely opaque in the visible; I think it's plausible. (That said, it's also very plausible that I've done the calculations wrong or misunderstood something; this is very not my field!)He can probably carry at least sixty pounds and still move, even without chakra boosting, which works out to three pounds per square foot. Extrapolating from https://www.eichrom.com/wp-content/...ttenuation-White-Paper-by-D-M-rev-6-1-002.pdf assuming gold works about as well as lead, and that we're doing initial experiments with UV and X-rays rather than skipping straight to gamma rays in the 500 keV and up range, I do actually think that would provide a useful amount of protection. Not long-term safety, sure - just to survive initial mishaps, collect enough clues about what's going on (e.g. "shadow clone survival time drops with remaining distance squared as they get closer without cover, but gold-plated shadow clone lasted five times longer than the one in a standard CHAOS suit at the same position") to start developing more thorough precautions in-character.
As @strange_person says, there's no obvious reason Hazō should have to jump straight to gamma rays; it would make perfect sense for him to try "past-past-blue" before "pastest-blue". In fact, this would provide the major advantage that a large enough thickness of stone (e.g. MEW) would probably work to shield him from it, and unlike the gold armour that's a thing that Hazōpilot would plausibly think of immediately ("well, I can't see the light, but if it's light maybe a wall will block it out").