All of the major international car hire firms are present in Bucharest, alongside a number of reliable local outfits. All have representations at Henri Coanda (Otopeni) airport, as well as a number of city centre locations (often in the five-star hotels).
You will usually need to be over 21 to hire a car in Romania, and to have held a license for at least couple of years. You will need a credit card, and non-EU citizens will need an international driving license.