Let's take the eponymous feminist cases first...
Care implies doing tasks for someone incapable of doing those tasks for themselves. Young children, older people with dementia, people with severe disabilities, etc, all require care, since someone needed to feed them, clean them, move them about, etc. Care can be given for necessities (food, shelter, etc) or for pleasures (walks in the park, providing entertainment or education, etc), but still provides something to the cared-for that they could not provide for themselves.
Service is providing a convenience for someone who could provide for themselves. A typical 1950s housewife would cook meals, wash clothes and linens, care for children, clean house, and do other activities that her husband was clearly perfectly capable of doing himself. Such a husband could hire someone to do all of these things if his wife went on strike; there are entire industries dedicated to providing such. That ability to hire someone is one of the hallmarks of a service.
There are of course, a number of confusing issues here. For instance (as noted above), a child requires care, but the act of providing care can be a service performed for another adult, so that said adult does not need to provide care for the child. A sick person might be perfect capable of getting up and (say) cooking themselves dinner, but someone might choose (or be paid) to provide that service, which we usually think of as caring for that sick person. Sex opens up an intellectual can of worms, because it is variously treated as an act of intimate mutual care and as a transactional service people provide for each other. All in all the care/service distinction is a sound typology with some decidedly fuzzy edges.
EDIT
I forgot to add (for completeness' sake) that while the concept of self-service is empty, the concept of self-care is useful. Self-care means doing for yourself what you wish another would do for you as an act of service or care. Taking the time to make yourself a home-cooked meal instead of picking up take-out would be an example of self-care; you are taking the time to do something for yourself rather than farming it out to a service industry. It's an idea worth exploring...