As several others have already said, don't give up! I wouldn't count yourself out by any means. Getting healthcare experience is going to help enormously, which it sounds like you're working toward.
Several others have already mentioned CNA experience, which is definitely awesome and translates well into an understanding of nursing. Personally, I went the route of becoming a Medical Assistant instead of a CNA. There are two types of MAs in Washington; Certified and Registered. Certified MAs go to school for about a year and a half, and pass a licensure exam. Registered MAs do not go to school and are trained and licensed by their employers (I.e., you don't take your license with you if you quit). I found an employer that trains people to become Registered MAs, and I've been working full time as a Medical Assistant & Phlebotomist for years. Going the CNA route is definitely more standard for folks seeking medical experience for the ABSN program, but the alternative route of becoming a Registered MA worked really well for me. A lot of Planned Parenthood clinics will hire for Registered MAs, and I've even seen some hospitals advertise those positions as "apprentice MAs". I also know at least one other person who went the Registered MA route and got accepted into the UW ABSN program.
All this to say, there are a lot of ways to get medical experience! Even volunteering in a medical context can really strengthen your application, and can give you connections to medical professionals that can write a letter of rec.
Also, as an aside, my Bachelor's was a BA and had nothing to do with science or healthcare. I studied feminist political economy. So it really is holistic!