Anthem Inc., the country’s second-biggest health insurer, said hackers broke into a database containing personal information for about 80 million of its customers and employees in what is likely to be the largest data breach disclosed by a health-care company.

Investigators are still determining the extent of the incursion, which was discovered last week, and Anthem said it is likely that “tens of millions” of records were stolen. The health insurer said the breach exposed names, birthdays, addresses and Social Security numbers...