Welcome to Burnham Rentals! NOW LEASING 1, 2, and 3 bedroom rentals for 2017-18. Burnham Rentals has several rental properties throughout Bloomington, IN with many locations within walking distance to Indiana University. Burnham Rentals is a family owned and operated leasing company started in 1942. Whether you're looking for an apartment or a rental home, Burnham Rentals has you covered! Call today!