I don't think you should get any errors if you don't store a class in a separate file in Processing, but it's still good practice, particularly in terms of code readability.

In order to do this simply create a new tab for the class in the processing IDE: each tab is saved as a separate file in the sketch folder.  Note that you can put any code into a separate tab (e.g. functions) and your sketch should still work just fine.  Also note that Processing isn't exactly the same as pure Java - any class you write is treated as a subclass of Processing, so not everything behaves as you might expect from Pure Java.

Some precisions on blindfish's good info:

• By "subclass of Processing", you must understand "nested class of the main Processing class" (which is named as the sketch name).
• When you create a new tab, by default (no extension) it creates a .pde file, so its content will just be added inside the main class.
• If you give a .java extension, it will compile the files without pre-processing them (no color types nor literals, for example), and the classes will not be nested. Side advantage: you can use the Java 1.5 syntax there (and double literals, etc.). Little inconvenience: beside the lack of pre-processing, you don't have direct access to Processing functions, you have to get a PApplet instance and use it.