Where does an elderly person with no money, only a small social security go when at the point they need 24 hour care?

This person needs to apply for Medicaid and enter a nursing home.

A nursing home, on Medicaid.

Medicaid will place her /him in a nursing home that will provide the services medicaid covers.
Usually these types of homes have a mixed population because they are private and run on a shoestring.
In other words the population may be young, old, suffering mental illness of all descriptions and disabilities of all types.
I suggest you do your own research. There are sites that list homes in all areas of the US with safety ratings etc.