I believe emergency services (ambulance/paramedic) fall to whichever county you happen to be travelling through, or the closest town. Not to scare you, but there are many stretches of trail where you will be several miles from the nearest town. Also, do not count on cell phone coverage - definitely bring one along, but coverage is spotty. If you are really concerned about this, pay close attention to the mile markers as you ride. If there should happen to be an emergency, knowing the mile marker number will make it easier to help authorities find you or your riding partner. Also carry a trail map or list of towns with mile marker numbers
- then at any given time you'll be able to determine whether the closest town is ahead of you or behind you. You'll also see the occasional farm house where you might be able to get emergency assistance, but not along the stretches of the trail beneath the river bluffs.
In addition to Hermann (as mcd mentioned), I know there are also hospitals in Clinton, Columbia, Jefferson City, Washington (near Dutzow), and St Charles.