Not far from Hyehwa Subway Station you can find the canal that stretches from Dongdaemun History and Culture Park all the way to the Koart Ville dormitory. Besides going for a walk and watching the sunset, this is probably the closest and most convenient place to go for a run for those of you who like jogging. For a longer run, however, I recommend taking the subway to the Han river. At Dongjak station you get directly to a nice park area that's perfect for a 10k+.
So far I have participated in two 10k races in Seoul and I will attempt a half marathon in May. It is a lot of fun and I can only recommend to get together with a group of people and sign up for the next race. Its a great experience to take back home. The races usually take place at the Han river and cost between 20.000-30.000 won (about $15-$25).