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Rub your hands with vinegar or lemon juice. For really stubborn stains, rub a mixture of salt and lemon juice on the stains.
I tried everything (vinegar, lemon juice, baking powder, and anything I could think of) to clean my hands of the nasty orange staining I got from peeling butternut squash. I was desperate because I didn't want to go to work with hands that looked absolutely awful.
On an impulse I washed with my mechanic son's GoJo (Orange Pumice Hand Cleaner). It worked wonderfully, even my nails were clean again. This would probably work with stains from beets, tomatoe, and cucumbers too.
As an aside, the pumice in the cleaner exfoliated my hands and left them feeling soft. I took the occasion to lotion them with Ponds Dry Skin lotion and wow, what a difference!
Source: Enlightening home experience