ANSWER: Are you feeling well?
I am from Texas, where we frequently say "hello" to strangers as we pass them on a sidewalk. In Texas, strangers even raise a hand in greeting as they pass each other in cars or trucks on narrow country roads. So, I was a pretty surprised when I first went to Korea back in the late 70's and discovered that Koreans did not greet strangers on the street, even if one of them was staring at the other as they passed.
Back in the late 70s and early 80s, foreigners were a much rarer sight in Korea than they are today, so Koreans would almost always stare at me when I would walk down a Korean sidewalk, even a sideway in Seoul. I remember the first time I said "안영하세요" to a Korean stranger on the street. He was a middle-aged man, who was staring a hole in me as he walked toward me on a lonely sidewalk. Since we had made eye contact, I felt obliged to say "안녕하세요" when he was close enough to hear it, but instead of returning the greeting or answering, he just looked a me as if I had asked if his wife were good bed. After a few more similar experiences, I learned not to greet strange Koreans on the street. By the way, even non-military foreigners back then would often ignore my greetings, as if they were New Yorkers or as if they had learned the Korean custom of not greeting strangers on the street.
Today, I read an article entitled "안녕" in Nam Yeong-sin's "Usage Handbook of the Korean Language" ("난영신의 한국어용법핸드북"). The first sentence in the article was as follows:
요즘 한국어를 배우는 외국인이 가장 먼저 익히는 말이 '안녕하세요.' ('하세요'의 높이가 거의 같도록 소리내.)라고 한다.
"These days, it is said that "안녕하세요" (saying '하세요' with almost the same pitch) is the very first expression foreigners studying Korean learn."
The greeting "안녕하세요" is supposed to be a question meaning, "Are you well?" but by saying that foreigners say the "하세요" part of it with almost the same pitch, Mr. Nam is suggesting that foreigners are not using the greeting as a question but rather as a simple "hello." He also suggests that even Koreans are forgetting the true meaning of the greeting. As an example, he gives the following anecdote:
두메 산촌에 의료 봉사를 하러 간 의대생이 거동이 심히 불편해 누워 있는 80대의 할아버지를 찾아가서 "안녕하세요, 할아버지?"라고 인사를 했더니 이 할아버지가 "아니, 난 안녕하지 못해요, 젊은이."라고 하더란다.
In a remote mountain village, a college medical student, who had gone there to do charity medical work, goes up to an old man in his 80s who was lying down because it was very uncomfortable for him to move and asked, "Are you feeling well, Grandfather?" The old man answered, "No, I'm unable to feel well, young man."
The old man's response was probably his way of saying, "Does it look like I'm feeling well? The point Mr. Nam was trying to make with that anecdote was that even some Koreans are forgetting the original meaning of "안녕하세요?" Mr. Nam did not mention it, but I think a more sincere and appropriate greeting to the sick old man would have been something similar to the following: "할아버지, 어디 아프세요?" which translates as "Grandfather, where does it hurt?"
I have not lived in Korea for about ten years, but I think today if a foreigner were to say "안녕하세요" to a Korean on a sidewalk, the Korean would probably return the greeting, knowing that the foreigner was not being nosey by asking about his or her health, but was just being friendly by saying "Hello."