What does "stable" and "reasonably accurate" mean in quantitative terms? The new PIC10F microcontrollers have built in 4MHz +/-1% RC oscillators, so that translates to a 1us instruction cycle. This means you could make a
21.739kHz clock, or a 22.222kHz clock. If you wanted to get hardcore and shoot for closer to your 22kHz you could play with tweaking the OSCCAL value for optimal 22kHz. Over the full temperature and voltage range (2.0V-5.5V) the intended frequency could be off by as much as +/-5%. These devices cost less than $0.70 in single unit quantities from Mouser and are available in SOT-23-6 or DIP8 packages.