There is no on-post housing for civilians on Osan Air Base. You will be living on the economy, which is very nice and and safe. Korea is listed in the top five safest countries in the world.
Just outside the main gate, there are several realtors that cater to base personnel. You can find a realtor on your own or get recommendations from co-workers after you arrive.
There are several residential areas in Songtan near the Osan AB that are suitable for teachers.
The first thing that you will want to think about is the type of home that you want to live in. Do you want to live in a high-rise apartment building, a smaller apartment complex called "villas" or in a western style stick-built house?
If you have children that may need to ride the school bus, please make sure to notify your realtors of this. They know the bus routes and will make sure to show you apartments or houses that are on a route.
Korea is a little different when it comes to paying your rent. In other areas you need to put down a deposit when you move into an apartment or house. This deposit is generally a month or two worth of rent. In Korea, you have the choice to pay your rent on a monthly basis, or you can do what most of the civilians do and pay your rent up front for the lease's length (usually one or two years).
Your utilities should also be included in your rent. Please be aware that you may have a cap on the amount of utilities that you can use. The landlord will average the amount you spend monthly. If you go over your monthly cap, you may be asked to pay for the overage out of pocket. Fortunately, most teachers leave during the hot summer month and don't need to use utility funds on air conditioning.
A Few Things To Be Aware Of
Electricity is extremely expensive in Korea. If you run your air conditioning constantly, expect to get $1,000+ electricity bills in the summer. Koreans traditionally only run air conditioning when they are at home.
You will have a difficult time finding apartments with 110v electrical outlets. Your apartment should come with some converters. If you need additional ones, please ask your realtor or ask a coworker where you can buy them inexpensively.
Your heat will most likely be floor heating. Please beware of over-using your floor heat in the winter as well. Gas can be just as costly as electricity.
The tap water in Korea is clean, but many Koreans choose not drink it unless it is filtered. If you choose to the drink tap water, please consider purchasing a water filter locally. Many people strictly drink bottled water while overseas. Water dispensers are inexpensive and 5 gal. water jugs can easily be delivered. (Ask your realtor)