You will struggle a little on second hand road bikes here. There are a few in some stores in Seoul but they are all high-end group sets that will be in the millions of won category. I will get a list of them however for you and post tomorrow.
Sells bikes and equipment online and I have found it to be okay for components, however as like all tech stuff in Korea a little on the pricy side. Have a look.
Sokcho has a bike store that sells road bikes. They are tube shifter bikes that come in around 250-300,000 won new. They are basic, but cycling is more about the motor in many cases. The store is just down from Hi Mart, and next to the door of AP. In addition, down Yangyang's main street there is a bike shop. He will order for you from a catalogue and would be worth a visit.
Depending on your high however, this may be a problem. The largest Korean frame I have seen here is 52CM, which is good for a 5' 8" person.
Neither stores speak English, but the Yangyang chat will be more accommodating than the Sokcho. None the less try both out and see how you go. I hope that has been of some help to you.