Anytime I've used OxiClean to clean a machine's parts, my nails have been the cleanest, clearest things you'd ever see.
Ok,.. the clearest part is more the "eggshell nails" I have, but previously, they would always be black from crud and oil and grease on the machines.
I sincerely doubt that Oxi is great for your skin (probably not worse than kerosene though, that really dries the hands and makes the skin feel awful), but if you were to use a scrub brush with a weak solution of that and Dawn, perhaps it would be OK once rinsed really well? Test on an area for any sort of irritation first, I guess, but it's never bothered my eczema.
ETA: The other thing is, a Car Detailer once told me "that which you ruin it with, you fix it with".... perhaps a greasy sort of hand cream,... or cleaner grease that you can wash off? The SuperLube that I've been trying on the machines lately is food safe, so I'd think it shouldn't irritate the skin too badly.