The best part about this potentially terrifying hike is that if you know what you're getting into, you can plan which terrifying bits you want to try, and which you don't. So make sure you do your research!
Mt. Dobong, located near Seoul (and accessible by subway...how crazy is that!), has a lot of ways up and a lot of ways down, but its almost universally known as one of the best hikes in Korea. It's also known as a mix of hiking and technically unchallenging rock climbing, but rock climbing nonetheless.
Here you can see that, depending on the route you take up, you may have to transverse some ridges using these big, well-spaced cables to hang on to. From everything I've read it seems like there's a lot of older (60s, 70s) Korean hikers that manage it just fine, so if you're brave enough, you can, too.
To avoid the route across the mountain ridge when going down the mountain (which would require you to use ropes and railings and lots of upper body strength), make sure to heed the signs. There will be a sign (with two little images of bears) posted where the two trails diverge, one trail will be marked with a RED line and the other trail will be marked with a GREEN line (the optimal route for the less experienced). Watch this video for clarification (or just fast forward the video to minute 2:33)!