The share of people under age 30 who own private businesses has reached a 24-year-low, according to new data, underscoring financial challenges and a low tolerance for risk among young Americans.

Roughly 3.6% of households headed by adults younger than 30 owned stakes in private companies, according to an analysis by The Wall Street Journal of recently released Federal Reserve data from 2013. That compares with 10.6% in 1989—when the central bank began collecting standard data on Americans’ incomes and net worth—and 6.1% in...