If you are willing to commit to some regular maintenance procedures, it’s entirely possible.
There is one caveat to my statement above before I discuss those procedures. If he is severely allergic to the point of an anaphylactic reaction which could be life threatening then that would not be wise.
The vast majority of people are not that bad so here are several things to do.
Be meticulous about vacuuming OFTEN.
The bedroom should be absoutely off limits to the cat at ALL times.
A hairless cat still produces dander. THIS is what people are allergic to. Not the hair.
Bathe the cat at least once a week and wipe it down all over with a damp washcloth. For some just the daily wipedown is sufficient. Experiment.
Get a good quality air filter.
Consult with an allergist regarding de-sensitization shots. If he tells you to get rid of the cat, find a more competent allergist.
All of these measures are designed to mitigate the total allergen load. For some people a few measures will suffice. For others, all of them need to be combined.
The main point is to keep down the level of dander floating around. This is what causes the symptoms. The dander is a protein substance which is produced when the cat grooms itself. The saliva dries on the coat and sloughs off in tiny particles. The less around in the environment, the less the exposure. Hence the cleaning, washing and air filtration.
Please consider fostering a cat temporarily rather than adopting one and then returning it. This lessens it’s chances for adoption and increases the possibility of it’s euthanasia.
Shelters welcome people willing to foster a cat temporarily. Many times it’s due to a cat recovering from an illness or a pregnant MOM with kittens. Don’t worry. She does all the work. All they need is a temporary space till they are old enough to be adopted. And if it works out for you, you could adopt one of the kittens.