Regional Governor Veniamin Kondratyev said that emergency services were working to put out the fire.
Local authorities said that the massive blaze engulfed several floors of a nine-storey apartment building and at least 17 apartments were affected. There was no immediate information about casualties.
The Defence Ministry said the plane that crashed had taken off for a training mission.
Roman Bublik, the head of the Yeysk district, wrote on his telegram channel: “An emergency has occurred in the city of Yeysk - a nine-storey building is on fire.
“All services promptly went to the scene of the fire. I am also at the scene of the emergency, coordinating the work of all services.
“A help desk is being installed. Residents will be provided with all the necessary support. The details of the incident are being specified. I will keep you all informed.”
Yeysk, a town of about 90,000, is home to a large Russian air base.
The Su-34 is a supersonic twin-engine bomber equipped with sophisticated sensors and weapons that has been a key strike component of the Russian air force. The aircraft has seen wide use during the war in Syria and the fighting in Ukraine.