Most host family accommodation situations are arranged through agencies that place students with a family willing to host them. In your case, you would have already graduated from high school by the time you'd be looking to come to Korea, so you wouldn't be a student anymore. Essentially what you're looking for is a free stay in Korea. You don't want to pay for a hotel, motel, hostel or other accommodation, so you're trying to network with a family that would put you up for free. Why would anybody be willing to do that? I mean, stop and think about it for a moment - why would anybody be willing to open their home to a stranger, a stranger from another country, from a different culture, a stranger that doesn't have proficiency in the local language to be able to communicate, why would anybody allow someone like that to come and live in their home for a week? The only way that anybody would be willing to do that would be if you were to make it worth their while. For example, there are some people in Korea who allow foreigners to stay with them in exchange for giving their kids English lessons every day. Do you really want to be obligated to spend an hour or two every day teaching English to a couple of Korean kids? I sure wouldn't.
South Korea is not that expensive if you plan your itinerary and budget well. You can get a ticket for under $2,000 round trip, you can book a room in a comfortable motel for anywhere between $50 and $100 a night, and a bed in a hostel could run you as little as $20 per night. You can find cheap meals here and get by on a few dollars a day. Bus and train tickets are very reasonable.
What I'm saying is that you shouldn't be planning a trip if you can't afford it. A week is not a lot of time. At the bare minimum, your trip is likely to cost you around $4,000, could be more, could be less, but after flights and accommodation and meals and transportation, that's a fair estimate for a full 7 days. The better you plan things, the more you could shave off of that. You might be able to get away with dropping only $3,000, but that would depend on how much preparation you put in before you arrive. Either way, you can't expect that a random Korean family is going to throw open their doors for some foreigner.
If you wan to come to this country, pay your own way. If you can't afford to do that, go elsewhere or stay home.