Yes it is possible to rent an apartment or home without proof of income.
However if you do have no proof of income then it can be hard to find a place to rent but not entirely impossible.
Just look for an apartment or home to rent where you consult with the landlord as an individual such as a home for rent by owner.
A home that is being rented by the owner will likely be a good place for someone to rent who has no proof of income.
But renting a nice apartment or a very nice house would require in most cases proof of income and good credit and good rental history as well.
When you apply to rent and apartment or rent a home and fill out a rental application you should hear back from the landlord or property management on the rental application within a few days.
However if the landlord or property manager is really busy then it may in rare cases take up to a week to hear back on the rental application whether or not you've been approved.
If you paid a rental application fee then it likely is not refundable as well and so you would have to pay another rental application fee to another landlord if you get denied the first rental.
You need to have verifiable income, good credit and good references to get approved easily on a rental application.
The landlord wants to make sure they get a good tenant who will pay their rent on time and who will take good care of the home or apartment they are renting