Lee Byung Ken stared at the guts of a Patek Philippe watch through a loupe held against his eye by a wirelike band.

“This had some water damage, so there was moisture below the bezel,” he said of the Nautilus Automatic model in rose gold with a diamond bezel. “Good thing the diamonds weren’t in danger.”

Another Patek, a chronograph, sat on a tool-cluttered table nearby — although nothing in his small 10-foot-square watch repair shop was very far away. “There was dust on the dial since it’s more than 10 years old,” he explained. “I had to disassemble it and then piece it back together. It’s like a puzzle.”

His fascination with clocks and watches started at age 16. “I lived next door to my uncle, who had a big mechanical desk clock, which I took apart and reassembled when he was on a trip,” said Mr. Lee, 76. “He never found out.”

A couple of years later, he attended a local watchmaking academy (which closed decades ago) and worked at a few of the watch repair shops around Seoul, mostly helping with mainspring repairs. “At that time, it was a very difficult to replace them, so I learned how to fix the bent springs,” he explained. “There was a big demand for this skill.”