CEDAR KEY, Florida — Idalia strengthened into a hurricane Tuesday and barreled toward Florida’s Gulf Coast as authorities warned residents of vulnerable areas to pack up and leave to escape the twin threats of high winds and devastating flooding.

Idalia was churning in the Gulf of Mexico as a Category 1 storm, but it was projected to come ashore early Wednesday as a Category 3 system with sustained winds of up to 120 mph in the lightly populated Big Bend region, where the Florida Panhandle curves into the peninsula. The result could be a big blow to a state still dealing with lingering damage from last year’s Hurricane Ian.

The National Weather Service at Tallahassee called Idalia “an unprecedented event” since no major hurricanes on record have ever passed through the bay abutting the Big Bend region.

On the island of Cedar Key, Commissioner Sue Colson joined other city officials in packing up documents and electronics at City Hall. 

She had a message for the almost 900 residents who were under mandatory orders to evacuate the island near the coast of the Big Bend region. More than a dozen state troopers went door to door warning residents that storm surge could rise as high as 15 feet.

“One word: Leave,” Ms. Colson said. “It’s not something to discuss.”

Not everyone was heeding the warning. Andy Bair, owner of the Island Hotel, said he intended to “babysit” his bed-and-breakfast, which predates the Civil War. 

The building has not flooded in the almost 20 years he has owned it, not even when Hurricane Hermine flooded the city in 2016.

“Being a caretaker of the oldest building in Cedar Key, I just feel kind of like I need to be here,” Mr. Bair said. “We’ve proven time and again that we’re not going to wash away. We may be a little uncomfortable for a couple of days, but we’ll be OK eventually.”

Tolls were waived on highways out of the danger area, shelters were open and hotels prepared to take in evacuees. More than 30,000 utility workers were gathering to make repairs as quickly as possible in the hurricane’s wake. About 5,500 National Guard troops were activated.

At Tarpon Springs, a coastal community northwest of Tampa, 60 patients were evacuated from a hospital out of concern that the system could bring a 7-foot storm surge.

“You do not have to leave the state. You don’t have to drive hundreds of miles,” Governor DeSantis said Tuesday morning at the state’s emergency operations center. 

“You have to get to higher ground in a safe structure,” he added. “You can ride the storm out there, then go back to your home.”

With a large stretch of Florida’s western coast at risk for storm surges and floods, evacuation notices were issued in 22 counties, with mandatory orders for some people in eight of those counties. 

Many of the notices were for low-lying and coastal areas and for people living in mobile and manufactured homes, recreational vehicles or boats, and for people who would be vulnerable in a power outage.

Many school districts along the Gulf Coast were to be closed through at least Wednesday. Several colleges and universities also closed, including the University of Florida at Gainesville. 

Florida State University at Tallahassee said its campus would be closed through Friday.