Ross Sapir’s customers are clear when they call to hire a moving van: They are leaving New York City because of the coronavirus crisis and they aren’t sure when, or if, they are returning.

Mr. Sapir, president of Roadway Moving, said his storage business is booming as people with the motivation and means flee the epicenter of the pandemic, which had killed nearly 11,000 in New York City as of Saturday. Many clients are leaving before figuring out where they are going and want movers to go into their now-uninhabited apartments,...