Without embalming, the body is kept cold. Dry ice packed under the torso, or reusable cooling packs, plus a cool room with the heat off.
How long that buys you depends on temperature, the person's condition at death, and the cause. Families commonly keep a body at home for one to three days. Some states set a limit on how long before burial, cremation, or refrigeration is required, so this is another point to check locally.[1]
What actually happens: the body cools and stiffens within hours, then the stiffness passes. There's usually no strong odor in the first day or two if the body is kept cold. Fluid can leak, which is why absorbent padding is standard.
None of this is dangerous to the people in the house. A body isn’t a public health hazard in ordinary circumstances, and that’s true whether or not the person had an infectious illness at death.[1]