I'm building a basic stereo system, using a car stereo and speakers, inside a large plastic box.
The stereo requires grounding, but the box is entirely plastic, so there is no metal chassis, and it needs to be portable, so I cannot connect it to an external ground/earth.
What should I use as a ground?
Portable electronics don't require users to connect wires to ground, so the literal definition of "ground" doesn't apply. In your device, "ground" means a common voltage reference, usually the negative terminal of your battery.
For an audio/amplifier, you may want to enclose portions of your circuit with shielding, which in turn is connected to "ground" (0V or negative battery terminal), to prevent external signals appearing in your audio output.
There are plenty of portable stereos with an outer shell made of plastic, but if you tear them apart you'll find some metal "boxes" surrounding portions of the internal circuitry, and this is to mitigate interference and help ensure proper function. Even if you don't shield your device, you will want to make sure that you've used ground throughout the device (ground plane on PCB, grounding straps to connect different boards, etc.). This will help reduce the electrical noise that your project emits.
If you're going to market and produce this, you'll need to make sure you conform to local regulations that pertain to devices accepting interference, and not generating more than an acceptable amount. (Though I infer from your question that this is a one-off.)
