Can a caregiver of an elderly family member get paid by Medicare?

No, Medicare will not pay for long-term care, either at home or in a facility. Medicaid will pay for long-term care if someone qualifies. In some instances there may be small amounts of money that can go to a family caregiver. If the person was in the service, the VA can also assist if the person has assets low enough to qualify. The application procedure for the VA can take quite a while, but it may be something you want to look into.