No injuries were reported.
The National Weather Service issued flash-flood warnings in 15 counties, saying record flooding was possible.
Water had reached the first floor of Johnson Memorial Hospital in Franklin, county Commissioner Tom Kite said, and cars were submerged up to their windshields in the county government building parking lot.
"We have dams failing in the Prince's Lakes area," threatening the town of Nineveh about 30 miles south of Indianapolis, Kite said.
The Indiana National Guard helped with rescues in Morgan County, southwest of Indianapolis.
Johnson and Morgan were among four counties in central Indiana where emergencies were declared, and shelters were set up for hundreds of residents forced from their homes by rising water.
Indiana State Police reported evacuations in the Lake Lemon area about 10 miles northeast of Bloomington. Dams near Gold Point were close to collapse, police said.
Interstate 70 was closed in Clay County in west central Indiana, and Interstate 65 and another major route, U.S. 31, both were closed near Franklin.
In western Indiana, water more than a foot deep surrounded homes on Terre Haute's east side.
J.D. Kesler, deputy director of the Vigo County Emergency Management Agency, said more than 200 people had to be rescued from their homes, vehicles and nursing homes there.
More than 30,000 electric customers lost power, the Indiana Utilities Regulatory Commission said.
Authorities also were cleaning up in northeast Indiana and northwest Ohio, where thunderstorms tore down trees and power lines Friday night.