It does sound eczematous and I do get on my hands, mainly my thumbs. I would just visit the pharmacist in the first instance who can give you some 1% Hydrocortisone to use for no more than 7 days and moisturisation is the key - as in keep applying as often as you can. The best moisturiser I find is the old fashioned Nivea in the round blue tub.
Being run down / stressed does exacerbate it for certain but since I've optimised my Vitamin D levels it has been a lot better for me.