Among those on board were 10 children, the nation's transport minister, Budi Karya Sumadi, told reporters.
The suspected crash site is near tourist islands just off the coast of Indonesia's sprawling capital.
Distraught relatives waited nervously for news at the city's airport. "I have four family members on the flight — my wife and three children," Yaman Zai said as he sobbed.
"[My wife] sent me a picture of the baby today … How could my heart not be torn into pieces?"
The plane took off on Saturday afternoon local time and a search and rescue operation began.
Data from FlightRadar24 said the plane reached an altitude of nearly 11,000 feet (3350 metres) before dropping suddenly to 250 feet. It then lost contact with air traffic control.
"Sriwijaya Air flight #SJ182 lost more than 10,000 feet of altitude in less than one minute, about 4 minutes after departure from Jakarta," the tracking agency said on its official Twitter account.
Broadcaster Kompas TV quoted local fishermen as saying they had found debris near islands just off the coast of the capital Jakarta, but it could not be immediately confirmed as having belonged to the missing jet.
Authorities and the airline gave no immediate indication as to why the plane suddenly went down.
But transport minister Sumadi said the jet appeared to deviate from its intended course just before it disappeared from radar.
Among the other passengers was Agus Minari and her husband, who were on their way back to Pontianak after visiting her son and attending a funeral in Java, according to her cousin Deni Triady.
"The family is deeply shocked," Triady added.
Russian President Vladimir Putin issued a statement offering his "sincere condolences" over the incident.
The budget airline, which has about 19 Boeing jets that fly to destinations in Indonesia and Southeast Asia, said only that it was investigating the loss of contact.
In October 2018, 189 people were killed when a Lion Air Boeing 737 Max jet slammed into the Java Sea about 12 minutes after takeoff from Jakarta on a routine one-hour flight.
That crash — and a subsequent fatal flight in Ethiopia — saw Boeing hit with $2.5 billion in fines over claims it defrauded regulators overseeing the 737 Max model, which was grounded worldwide after the two deadly crashes.
The Boeing jet thought to have crashed on Saturday is not a Max model and was 26 years old, according to authorities.
"We are aware of media reports from Jakarta, and are closely monitoring the situation," the US-based planemaker said in a statement.
"We are working to gather more information."
A search and rescue operation quickly began and continues today.
"We deployed our team, boats and sea riders to the location suspected to be where it went down after losing contact," Bambang Suryo Aji, a senior official at the agency, told reporters after nightfall.
Indonesia's aviation sector has long suffered from a reputation for poor safety, and its airlines were once banned from entering US and European airspace.
In 2014, an AirAsia plane crashed with the loss of 162 lives. Domestic investigators' final report on the AirAsia crash showed a chronically faulty component in a rudder control system, poor maintenance and the pilots' inadequate response were major factors in what was supposed to be a routine flight from the Indonesian city of Surabaya to Singapore.
A year later, in 2015, more than 140 people, including people on the ground, were killed when a military plane crashed shortly after takeoff in Medan on Sumatra island.
