The bodies of Richard Smith, 37, and wife Clair, 36, together with their sons Ben, nine, and Aaron, one, were discovered at a house in Pudsey, near Leeds.
Emergency services found the bodies in an upstairs bedroom of the property in a quiet cul-de-sac yesterday afternoon.
Detectives are working on the theory that it was a case of murder-suicide. No one else is being sought.
The family are thought to have moved into the modern stone-built house a few months ago.
One woman who laid flowers at the police cordon said her children went to school with Ben.
She said: "It's terrible. From what I saw of them, they were very fun-loving, kind and caring – full of life."
The woman, who did not want to be named, said: "They were a nice family." Another neighbour, who lives round the corner, said: "It's such a shock. We've been up all night watching what's been going on. "This really is a quiet street. You just don't expect it. "I didn't know them, apart from seeing them around and about. "They had two boys. It's just so sad."
Police and scientific support officers were still at the Smith family's home today.
The house, which has an integral garage, has a broken lounge window – the only other sign of anything out of the ordinary. Family photos could be seen inside. Two cars were parked outside.
A police spokesman said officers were called to the house at about 5.30pm yesterday.
Part of Sheridan Way, where the house is located, was initially cordoned off amid fears that dangerous chemicals might be present.
Detective Superintendent Paul Taylor of the West Yorkshire Police homicide and major inquiry team said: "Initially, we had concerns regarding the release of harmful chemicals or gases. "Following testing, we no longer have any concerns with regards to those working at the scene and we will continue our investigations today. "Inquiries are at an early stage and I would ask anyone with information about the incident to contact the homicide and major inquiry team."
The tragedy comes just days after a separate incident in Melton Mowbray when a police officer who had been dismissed by his force attacked his family before apparently killing himself.