In the 70s, my grandpa taught me to clean my filthy greasy hands with clean motor oil. He was born in 1896 and was a mechanic for the county highway department until he retired in about 1965 or so.
Aaaaannnnnyyyyyway..... I actually did that for a while in the 70s and 80s then follow with dish soap. Dish soap works well alone too, if you don't have a real hand cleaner - like Go-Jo or any of the commonly available citrus hand cleaners (which is what I use now). I often use Nivea, Eucerin or other fairly thick/substantial hand creams to get petro-stink off my hands if necessary because they're around the house.
Olive oil or vegetable oil? Never crossed my mind. I'd just use dish soap.