Well, I do wear gloves, almost all the time when woodworking.

Ordinary tight fitting thin work gloves (I get them in Germany as the brand I use lasts well and the rubberising lasts) and sometimes, when turning for example, I wear surgical gloves in a fetching shade of blue.
I've had the triumvirate of asthma, eczema and near year round hayfever and dust allergy since birth. I'm allergic to wood tar of all things (also found in paper) as well as weird stuff like cobalt and most animal dander. Very severe allergic reaction (ie go to hospital right now) to ingesting walnuts and walnut oil.
Any prolonged skin contact with most woods, results in eczema, usually at the tips of my fingers, which them crack and eventually bleed. Blood on wood is not a great look :lol:
This is possibly not ideal for having woodwork as a hobby, but you can't let this stuff stop you doing things. As my main pastime now is music - piano a lot and guitar a bit, it stops me totally if the ends of my fingers are cracked. Blood on the keys and wet strings! I can cope with that but it hurts too, and makes it hard to focus.
So I wear gloves when woodworking, or painting, or fixing mucky machinery. Its totally fine, I am super careful and I have never had a glove related accident. (Famous last words). Just about the only thing that poses a real risk is turning, which I do occasionally when I need a spindle or something. Then I wear surgical gloves, which I am confident will rip long before I am dragged into chucky the man mangler.
Eczema is a weird thing. I keep it entirely at bay most of the time with a combination of antihistamines, moisturiser slathered on and sometimes cortisone (steriodal) creams. However, having creams on the skin can leave oily residues on things like pristine white maple, or even plywood. Odd things can trigger allergies though - for example near to Christmas time I usually have an allergic reaction to apples, and sometimes peas. I really like peas.
I blame my father, which is a variation on the trope "if it's not one thing, it's your mother". My son has inherited this nuisance as well. He is more keen on getting elbows deep in engine oil, rather than sawdust though.