Is it possible for a green card holder to rent a house without proof of income/job?

This is not really a legal issue, but a business issue. If she can convince the landlord that she can pay the rent, she may be able to rent an apartment. Her immigration status does not matter.

The landlords are not required to accept anyone, be it a US citizen or an immigrant. It is up to a landlord to determine what they are comfortable with. The apartment complexes usually have their procedures in place, they do a background check and employment history. Some private home owners may be satisfied with a less rigorous check. But if they are smart, they will want to make sure that your friend can afford the rent. No one wants to have to evict a tenant if she does not pay rent. But if a person does not have any income, my guess is, she will have a hard time finding a landlord who will agree to rent her a place (and take a risk that she will run out of money and will not pay). Of course, she can agree to pre-pay several months of her rent, and that might make a landlord more comfortable, but there is no requirement that a landlord must accept a green card holder regardless of whether she is working or not.

It is entirely up to the landlord. There is no involvement with the government. Many landlords rent to people without the right to stay permanently in the US. Your friends problem is that without a job or source of income a landlord is not likely to rent to anyone.

This is not an immigration law problem. Anyone can rent an apartment, if they qualify to the satisfaction of the landlord or property owner. Without a job or proof of income, very few, if any, landlords would rent an apartment to a new tenant.

Her green card does not matter.. if a landlord will accept her without income he will.

I would suspect that the green card holder would be able to rent a house as long as she satisfies the landlord that she is able and will pay the rent.

I am not an immigration law attorney. However, landlords are entitled to assurances that the rent will be paid on time and that their renter has sufficient income or funds to pay for any damages which may be caused by the renters or their family. The $5,000.00 may be sufficient but the money wouldn't be available for anything else.