Drying Apricots in the Oven. 1  1. Buy your apricots when they are fully ripe.Under-ripe fruit can become sour when dried. 2  2. Look for big sales at your local supermarket. 3  3. Ripen firm apricots by placing them in a paper bag in the window.  4. Clean your apricots 1  thoroughly. 5. Pit the apricots.  6. Turn the apricots inside out.