What happens to an elderly person who has no retirement savings, no job or money, and cannot stay with their relatives? Where do they go?

I just read your other post and answered there too.

https://www.agingcare.com/questions/what-do-i-do-493588.htm

Does this person have dementia? or other profound health issue? If so you report them to APS (part of social services) for their county. They will get emergency guardianship powers to remove them to a facility if they are an immediate danger to themselves or others. A combination of Medicaid and their SS income will pay for the facility. If not, then look into section 8 housing and financial aid that may supplement their SS income (if there is such a thing).

If living in the US and worked 10 yrs, they should have Social Security and Medicare.

I would call Adult Protection Services and let them take over. They can help them find resources to help them live on their own. If the person needs 24/7 care, they can get them into a nursing facility. If you are a relative, APS may try to get you to care for this person. Be firm telling them you are not an option. If not related then your just a person trying to get them help.