Depends on how nice you want the place to be and how big/how many bedrooms. If you are fine with a bare minimum of furniture and don't mind the place being small or not very well kept, you might be able to find somewhere. You could also look for unfurnished apartments and negotiate with the owner to furnish it for you.
My place, including utilities and furnished, if I were to rent it out (it's a modern condo with central AC, granite etc) would be at least $1300.
Try advertising on craigslist yourself, under "housing wanted." Good luck.