The bill would add pressure on social networking services such as Twitter and Facebook, which do not run servers in Russia.
The law will mean that from 2016, all Internet companies will have to move Russian data onto servers based in Russia or face being blocked from the web.
The bill must still be approved by the upper chamber and President Vladimir Putin before it becomes law.
Other Russian laws require blogs attracting more than 3,000 daily visits to register with a communications watchdog. Another one is directed against "extremist" language that could see Russians go to jail for up to five years for retweeting offensive information. Websites in the country could be also shut without a court order.