The job market is improving, but one statistic presents a stark reminder of the challenges that remain: Nearly half of the unemployed—45.9%—have been out of work longer than six months, more than at any time since the Labor Department began keeping track in 1948.

Even in the worst months of the early 1980s, when the jobless rate topped 10% for months on end, only about one in four of the unemployed was out of work for more than six months.

Overall,...