Kenmore water heaters can be somewhat of a pain to make repairs on. However the most common issue with Kenmore water heaters is actually a very easy fix. The Kenmore units have an air intake filter that can often get clogged with dust or cobwebs. A clogged intake will cause the pilot to go out.
There are two types of Kenmore water heaters, both are pictured below. The first image shows a Kenmore unit that has three legs. This type of unit needs to be installed on bricks in order to receive adequate airflow. The bricks also help provide access to the filter in order for yearly cleaning. If the water heater was not installed on bricks, you must drain down the tank and sit on bricks. Once you have access to the bottom, use a shop vac to clean the intake.
The picture shown below is a more common type of Kenmore and does not need to be installed on bricks. The filter comes off very easily and can be cleaned out with a shop vac or sometimes just blow out manually.
Start with cleaning out the filter and if the pilot still won’t stay lit you may need to replace the thermocouple. The thermocouple will usually shutoff the pilot light after about 20 seconds and you should hear a click. This is a safety device that turns the gas off when the pilot goes out. Most Kenmore units have an electronic or fused thermocouple that must be purchased from Sears. In some cases they may not have them in stock.
When you replace the thermocouple on a Kenmore water heater you should also replace the combustion chamber gaskets in order to provide an adequate seal. The gaskets can also be purchased from Kenmore.