I use oil & lanolin.
Constant scrubbing with soap strips the natural oils from your hands. You can replace this protective layer with a neutral vegetable oil (I like almond oil, and have also used sunflower & olive oils) just rub it into your hands well - good for nails too. Do this whenever you don't need to be handling food or washing hands - just before you go to bed is a good time. Lanolin is another good simple skin-protecting product (sheep shearers always have baby-soft hands, and they eventually figured out that it's due to the lanolin on the wool), but is very sticky and some people dislike the smell. Lanolin is the base of many many skin creams, lip balms & "scarless healing" creams.
I don't know what kind of soap you're using, but you may be able to find something less harsh on your skin. For everyday hand & skin washing I usually just use sorbolene cream, but I don't know if this would be good enough for food-grade sanitation.
Or yes, you could wear gloves - get powder-free ones though.
But I agree that you have to do something about this - if you've irritated your hands to the point where they are almost bleeding (are they cracked?), then that is a hygiene problem in itself.