Most cars that look good, have low maintenance requirements and are safe are close to impossible to find for under $5,000. So safety and low maintenance are probably going to come first and make the most sense when buying a car of this caliber. Safety itself, when it comes to buying a car, can be calibrated in a variety of different ways, based on past crash tests, size and weight of the car itself and if it has any airbags, assuming they were installed whenever the car was made. These are all important factors to be taken into account before laying down that $5,000 (or less).
Carfax.com supplies many forms of information about automobiles of all types, but can be specifically used to scout safety and reliability ratings of used and older model cars dating all the way back to 1991. A specific model can be searched by year, make and VIN number if you have that available for the specific car you are interested in purchasing.
"Consumer Reports" can provide similar information as Carfax.com, reviewing all makes and models of automobiles as well as tips on driver safety in general. Check out its used car buying guide for more specific things to look for or maybe cross check the information from "Consumer Reports" with what is discovered from Carfax.com. This site will also include information on recalls or repeated accidents or incidents that these types automobiles were involved in.
Known as one of the safest cars on the market for decades, an older model Volvo even without airbags is one of the safer cars on the road. The makers of Volvo, even before merging with Ford Motor Company, would consistently build solid, strong automobile bodies that would withstand collisions and not collapse on the passengers inside. Volvos are also reliable and run considerably well and consistently.
Surprisingly, one of the safer cars on the road is the Saab 900, though they haven't had the best record for keeping maintenance costs low in their older models. Saabs have sturdy bodies and can take a hard impact quite well, protecting driver and passengers alike.
The Audi (from the early 1990s on), BMW 5 Series (from 1988 on) and the Mercedes E-Class (from 1986 on) are all safe and probably reasonably cheap, depending on miles, condition and how much the seller loves the car. They are big bodied German cars that get horrible gas mileage, but will keep their shape and protect the passengers and driver in an accident, assuming everyone is wearing their seatbelts.