Prepare marinade with the orange juice, vinegar, achiote, garlic, and salt, to taste, and mix thoroughly. Rub the mixture into the meat and marinate at room temperature for 1 hour, or refrigerated, for at least 1 hour.
Preheat oven to 300 degrees F.
Line a roaster pan with heavy foil and arrange the marinated pork in it, covering the meat with the marinade. Cover pan with foil tightly and bake for 2 to 3 hours. Check often and turn pieces of meat in their own juices. Roasting times vary according to the toughness of the meat.
Chop meat into cubes and discard any fatty parts. Serve with a basket of freshly made corn tortillas.