How do I make a remote control car without LEGO motors?
It sounds like you are asking how to motorize your own custom LEGO creations using motors that you may already have around the house rather than purchasing official motors.

Before I answer this, I just want to make sure that you are aware that you can purchase the PF motors individually on LEGO.com for fairly cheap. You don't have to buy large, expensive sets in order to get motors. The basic PF motor set which includes a battery box, motor, wiring, and lighting is $30 USD. Unfortunately, as pointed out in the comments, this kit doesn't include a proper RC remote, so you'd need to buy an IR receiver and IR remote as well, which would add another $25 to the cost.

The first thing that you'll need to do is find a suitable motor. Old toys are probably your best bet. If you're going for small, you could also take a pager motor out of an old cell phone. This is what Phillipe used to make a motorized city-scale vehicle in another question:



The main challenges with using custom motors are:

Attaching the motors to LEGO bricks

The strongest connection will likely be created using some sort of glue or epoxy. Most two-part epoxies should be sufficient for small motors. Here's an example from Eurobricks:



You could probably get away with using screws or even rubber bands, but it's a bit trickier to make everything line up the way you want.

Converting the output of the motor to rotation of a LEGO axle

Assuming that the motor axle is relatively small, it is possible to drill out the center of a Technic axle and slide it onto the motor's axle. This can be a little tricky to get right, but it does work. Here's a video that explains it.





If your motor already has a pulley attached, it's easy to just run a belt between the motor and a Technic pulley.

Powering and controlling the motors

If you don't mind your creation having a tail, the easiest way to power the motors is to simply run a wire to the vehicle from some sort of controller. It's fairly easy to put together a working system if you have old wired remote controlled toys like this one laying around:



If you have any radio controlled toys laying around, it's also fairly easy to just rip the electrical parts out, put those on your LEGO model, and wire up your motors. You could also use the motors from the original toy. Then you don't even need to do any wiring. You would have to find a place for batteries and the receiver hardware on your creation, though.