ANSWER: A "barley fart" (보리방구/보리방귀).
Koreans apparently used to believe that the more barley you ate the more you farted. They also apparently believed that your farts smelled worse if you ate rice with barley mixed in it. So Korean school kids in the past probably wouldn't have wanted to share a 2-person school desk with a poor kid who ate rice with a lot of barley in it, especially if that kid rarely washed his hair and had body odor that smelled like a "cesspool" (시궁창). In Korea, poor people used to mix barley in with their rice because barley was cheaper than rice.
Now imagine that your teacher makes you and the students in your class change "desk partners" (짝꿍) once every month. Last month the girls got to choose the boy they wanted to sit next to in class, and this month the boys are supposed to choose the girl they want to sit next to. The "barley-farter" (보리방쥐쟁이) in this class is 조수택, who works after class delivering the "evening newspaper" ( 석간신문). Which girl will he choose to sit next to, and how will that girl react?
In the past, when 조수택, who is better known as "보리방구", was assigned a new desk partner, the new partner would always ask the teacher for a different partner, but the teacher always refused until 조수택, sensing that the student didn't want to be his desk partner, asked the teacher himself if he could sit alone in the back because it would help him study better since it would give him more room to hang his bag on his chair and more room to spread out his elbows on his desk.
So, which girl in class will 조수택 choose to sit next to, and how will that girl react? The only hint that I will give you is that there are a lot of sad stories in Korea.
By the way, the adjective 멋쩍다 means "to feel embarrassed, awkward, or uncomfortable," but 멋 by itself can mean "taste," "charm," or "elegance," and "-쩍다" means "to feel" or "to have a feeling," so why does 멋쩍다 mean "to feel embarrassed or awkward" instead of "to feel charming or elegant"? I am not sure, but maybe the word 쩍다 here is a dialectical pronunciation for 적다, which can mean "few" or "little." So, maybe 멋쩍다 literally means 멋이 적다, which could translate as "lacking charm or elegance," and that would imply "awkwardness."
