Depending on where you live, it is indeed possible to buy an old police car. Most police forces around the world are constantly in the process of updating their vehicle fleets, and they typically sell off old vehicles through government surplus auctions. As a general rule, old police cars are more readily available for sale in urban areas, where a large fleet of vehicles is necessary and ample funding allows for frequent updates. In isolated rural areas, it can be more difficult to obtain a police car, especially if you live in a region where police are underfunded.
There are several ways in which you can buy an old police car. Many police forces hold periodic auctions in which surplus equipment is sold, typically at a steep discount. Large police departments may have their own auctions, while others may bundle their goods in with a larger government surplus auction which combines equipment from multiple agencies. Old police cars are sometimes also sold at used car sales and auctions, especially those held by government agencies. These auctions are all publicly listed in the paper of record for the region. Sometimes you can also buy an old police car directly from the department, especially in a small town, in which case you may want to ask the department staff about the possibility of buying retired police cars.
If you buy an old police car at auction, you are usually not allowed to test drive it beforehand. It's a good idea to assess the condition of an old police car as best you can before buying it, because police officers are very hard on their cars. If you don't have a lot of car repair and mechanics experience, find a friend who does, and ask the friend to come with you to the car lot to inspect the car and look for obvious potential problems.
Once you purchase an old police car, you are required to remove identifiable markings so that people on the road do not confuse you with a real police officer. Some police departments paint their cars with primer before sale to ensure that they are not mixed up with the real vehicle fleet, while others simply ask that people remove department markings before driving. People who buy vintage police cars may be allowed to store the original markings, due to the fact that the age of the car clearly differentiates it from the rest of the fleet.
The lights and siren are usually disabled before purchase as well, so that they are not abused by civilian owners. You will also usually be required to get new license plates, since police cars are often identified with government plates, so when you register your vehicle, make sure to specify that it is a decommissioned police car so that new plates will be issued for you.