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Stainless steel will still need to be seasoned or **** will stick like crazy. It just won't rust if you don't season it right.
Wash the old carbon steel wok really good. Use 90% alcohol to remove all the oil residue and use a Brillo pad. Put your elbow into it man!
Then on the burner, use peanut oil or lard to season the wok. You heat the oil on high until it just starts to smoke, then let it cool and wipe it with a peanut oil soaked towel. Repeat the oiling, heating, cooling, oiling process 5-6 times. It will eventually take on the nice dark brown caramelized color.
Then when you cook in it, you heat it on high until it just starts to smoke again, dump the oil in and cook.
When done, you just wipe it out, then use hot water to rinse it out, a mild plastic bristle brush like a potato scrubber if you have tough stains, then put it back on the burner on high to dry it. If has areas that are shiny, wipe them with oil, then heat it again and let it cool. Then put it away. I have a carbon steel flat bottom wok and a carbon steel round bottom wok for the grill's gas burner. Both work great (though I use the flat bottom on the most powerful Jenn-air burner) and I've never had a problem with feeling sick or chipping seasoning.
It sounds like a lot of work or a PITA, but it's really not, it's quick and and easy clean-up once it's seasoned. In the summer the round bottom gets used about twice a month as we eat mostly raw veggies and cold dinners, but I the fall and winter, the flat bottom is used 2-3 times a week. Mine are stored in the basement too and rust has never been an issue.
I only use bamboo utensils in mine.
EDIT: The initial "seasoning" is just to get it prepped for cooking the first time, it takes years of use to really become a seasoned wok. My flat bottom wok is just starting look like a restaurant wok in color if that helps.