Anyone can buy a car. You'll need cash, your passport and a CDI (it's a "tax" ID code given to foreigners without residency when doing something that would normally require a worker's tax ID). You can go to an ANSES location (for your area) and get one pretty easily.
You also have to insure the car, which can be a little complicated, but not too much. Some companies will, and some won't, insure from a passport. I have Meridional, which when I bought my car some five years ago accepted just passports.
I've posted about buying a car before, as have others. I described why I'd buy a car from a used car lot as opposed to directly from the owner, mostly for paperwork reasons related to getting the car and title registered and the Cedula Verde (the registration card) in your name, etc. Traffic violations go with the car and the car can't be sold until all of that is cleared up, which the dealerships deal with as part of the whole process (when they bought the car to begin with, for example).
I bought my car from Centro de Motores somewhere around the 3000 block of Cordoba. It's a Renault dealership. They have a new and used car lot, across the street from each other. The used lot, when I bought mine, had a number of different makes available. I bought a 1999 Renault Megane hatchback. It was expensive - cars hold their value here.