While there may not be any such thing as a cheap classic car, there are certainly models that are less expensive than others. Start choosing the best one for you by deciding how much work you are willing to put into restoring the car, as well as from what era you want the cheap classic car to be. A car that is in need of a significant amount of refurbishing, or a complete rebuild, is likely to be far less expensive than a classic car in pristine or even working condition, and reproductions are less expensive than authentic classic cars.
It helps to begin doing research on the different types of cheap classic car models available. Generally speaking, the more highly sought-after the car is, the more likely the price is to be quite high. Look for a cheap classic car that is less popular, though still unique in many ways. Your goals for owning the car will play a role in choosing as well, since many people choose classic cars for the experience of refurbishing them. Others simply enjoy driving an older vehicle with a history. Decide why you want a classic car in the first place, and then do research to find cars that will fit your criteria.
If you want the look of a classic car but do not necessarily need an original, consider a reproduction or a kit. This will give you a cheap classic car, and while it is not authentic, it will give you the visual appeal and driving experience you are looking for. Reproductions may already be built and ready to drive, while kits will need some assembly. If you want to build the car from the ground up, a kit is a great way to get a classic car without spending a significant amount of money.
You may be able to go to a junkyard or classic car lot to find a cheap classic car in need of major refurbishing. The car is likely to be missing key components, it may not be in working condition, and the body may be damaged. This can be a fun project, but keep in mind that you will end up spending more money on parts, and you will spend a significant amount of time on the project itself. A project like this can take several years to complete in most cases.