Repeatedly exposing your hands to noxious chemicals should be avoided whenever possible. Certainly ,adjusting the load in your washer does not require a trip to the ER but you should be aware that the palms of your hands and the bottoms of your feet allow chemicals to be absorbed directly into your blood stream. Other sensitive areas where blood comes close to the surface of your skin also allow for osmosis : transfer of substances into the blood stream.
It is not bad to get a little bleach on your hands however prolonged exposure can cause severe skin irritations. In addition, if it is chlorine based bleach it can also cause painful chemical burns if you are exposed to it for long.
Some people can not breathe the fumes from bleach. When they do they can't breath. Diabetics should rinse their white clothes that have been bleach twice to get all the bleach out as it will irritate the skin where the clothing rubs like sock tops especially. Bleach can leave red blotches on the face that will peel like a sun burn. Bleach dries the skin. It's important to wash and rinse well after being exposed it. Dry chlorine bleach fo hot tubs and pools will do skin damage also.
It doesn't , until you have a longtime exposure to then you should be worried. But if your just washing a shirt with bleach than its ok as long as you wash your hand thoroughly with soap and warm water for like 30 sec.