Since many people come to Qatar on business, renting a car in Doha is a popular service. There are branches of large international rental firms and small local agencies, where the price of a rental car in Doha will be lower for a car rental in Doha.
Renting a car is easy enough, the only catch is that young people under the age of 21 can't rent a car in Qatar. If you are 21 years old, you can't rent a car in Qatar. If you are of a suitable age, you'll need to show your driver's license and you're good to go.
It's important to note that you can drive with your license (even an international one) for no more than 7 days. You can only drive for a maximum of 7 days with your license (even an international license). After one week, you must get a Qatari provisional license.
Renting a car in Doha without a deposit is possible if you are paying with a credit card. Many agencies will accept payment in cash, but you will need to leave your passport and a deposit (usually 300 USD).
Cheap car rental in Doha is possible if you need a "workhorse" and a simple car. It's worth noting that it's not uncommon to rent expensive executive cars in Doha, often with a driver.
Car rentals in Doha are available 24 hours a day from most of the major agencies, including airport rental shops - here, however, the price will be higher.