What can you do when you have an elderly parent that is physically nor mentally able to stay in their own home and does not want to stay with anyone either?

Really, there are only a few choices if they do not want to live with family: assisted living/nursing homes or in-home care.
The tough part is getting them to come to that realization. Without buy-in from them, it is so difficult to even begin to make plans.
Have a frank discussion when the time is right. Gather information about nearby facilities and then give your parent a long while to think about it so that it becomes his/her idea.
It is so unfair for parents to tie your hands because they think that they can stay in their homes. As children, we were not allowed to do things that our parents deemed unsafe. Why should the rules change when we become caregiver?
good luck

When my father-in-law died, my mother-in-law lived in that little house by herself for about a year. She has macular degeneration and at the time her memory took a jolt after his death and threw her into dementia quick. She started falling at home and not telling me. I would come and pick her up just to find her rubbing her head. When I'd ask her what the problem was, she said it hurt. Well, she had a big goose egg where she'd fallen, hit her head, and couldn't remember why her head hurt. She fell and broke her foot, so I had an emergency call system put in, so she had a button that hung around her neck all the time. That came in handy when during her last fall she broke her hip. Thank goodness she remembered the panic button. While she was at rehab all the doctors that treated her did an evaluation of her mental/physical health. They unanimously agreed she could no longer live at home alone. My point I guess is, start with a doctor who will evaluate them and tell you if he agrees that they can't live alone anymore. Be prepared, it's gonna be bad. Steel yourself for the kicking and screaming part, cause it will happen to a certain degree. Do what you have to do. We had to make the hard choice for my mother-in-law, but it was pretty bad at first. In her mind she knows she couldn't live alone anymore, but in her heart she wanted to go home.