DAEGU, South Korea—Cafes here demanded orders must be takeout or delivery. A typically bustling market hollowed out. The rare flicker of activity occurred at stores selling face masks, though most had run out.

These are the scenes of eerie silence unfolding in Daegu, South Korea’s fourth-largest city and an epicenter for a coronavirus outbreak that skyrocketed to 763 cases on Monday morning—a roughly 25-fold rise in just five days. In response, President Moon Jae-in raised the country’s virus-alert system to the highest of...