A couple days a week I have meetings in the busy Gangnam area and the meetings finish just before lunch. After the meeting, I am in the thick of the lunch crowds- all hurrying in groups to get a meal. Dining alone in Korea is almost unheard of (especially if you work at a company). The corporate work culture is: you eat with your coworkers and then you get coffee with your coworkers and then you go back to your office with your coworkers. Oh, and then after work you go drinking with your coworkers.
Geez, I don't know if I like any of my coworkers to spend every second with them. I mean I need some "me" time and I'd like to spend lunch alone from time to time. Also, sometimes I don't want to eat what everyone else is eating. Eating hot soup and rice everyday- although healthy- makes me sleepy and sometimes I just want a big salad or just fruit. Sometimes, I don't eat lunch at all.
Anyway, dining alone in Korea will draw stares from others that enter the restaurant. They might assume that you are an outsider (a wangtta) or you don't have a job. But most often, it might be seen as rude for you might take up a table for 2 or 4 by yourself and you eat all the side dishes (banchan) and main course by yourself.
What I've found is that there are ways around this. Most western eating establishments such as Italian, Mexican, Indian, Japanese, etc don't tend to mind people eating by themselves. Also, noodle restaurants don't seem to mind so much either. The other important thing is time.
If you are eating along go early like 11:30am or go at the tail end of lunch-12:45 or 1pm. At this point there are tables empty and people don't really mind. Now in the busy districts of Seoul, more companies are taking later lunches such as at 1pm. They will often reserve whole sections for their companies. You might want to avoid those.
Other places that don't mind you eating alone are fast food or franchise places such as Kimbap Chunguk, Kimbap Nara, or O-mori Chiggae. Small mom and pop 분식 (boon shiks) don't seem to mind all that much either if they are not busy. Oh, Korean restaurants that are foreigner friendly don't mind either.
The last places where you can easily dine alone are at bus stations, train stations, and other places of transit.
Overall, you won't have a problem dining alone. If a restaurant is very busy, they might ask you to move. Just out of courtesy to the restaurant, I would do it. They aren't trying to be rude, it's just that the restaurants are small and they have to make use of every inch possible.