Jeju Island is the gem of South Korea and now a UNESCO World Heritage List. If there is a more beautiful journey in Korea, I challenge you by asking you to compare it with this one - cycling around Jeju. You can start anywhere along the coast and simply keep cycling along the water. Whenever you see a sign for the beach road, follow it. The total length of the journey is around 250 km, although you can reduce it by skipping some of the beach roads and cycling along the 1132 instead - however, be highly aware of doing this near Jeju city, where traffic is high and drivers have little respect for bicycles - everywhere else, you will be a happy cyclist with flies upon your teeth and free to swim in the warm, clear waters off the coast of this iconic volcanic island.
Note: The 250 km ride is pleasant, solid riding for 2 full days on a semi-decent bike, although you can easily do it slower... or quicker.