Pet allergy is usually a reaction to proteins in an animal's dander (skin flakes), saliva, or urine rather than the fur itself, which is why no breed is truly hypoallergenic. For many people with mild to moderate symptoms, it is possible to keep a pet while reducing exposure rather than rehoming the animal.
Helpful measures include keeping the pet out of bedrooms and off soft furnishings, washing hands after contact, vacuuming and cleaning frequently with a filtered cleaner, using a HEPA air purifier, and grooming the pet regularly (ideally done by someone without the allergy). Hard floors are easier to keep allergen-free than carpets. These steps lower, but do not eliminate, the amount of allergen in the home.
Medicines such as antihistamines, steroid nasal sprays, and eye drops can control day-to-day symptoms. If the allergy is severe, or if it triggers asthma with wheezing and breathlessness, the most effective step may be to avoid the animal, and a doctor can also discuss whether allergy immunotherapy is appropriate. See a doctor if symptoms are hard to control or affect your breathing.