When the Stihl 029 chainsaw runs, it needs to breathe just like a human. It also needs fuel to energise this process. If the process is interrupted, even for a short time, your engine will shut itself off. So if the Stihl starts, runs for a bit and then dies, you likely have a problem with either the airflow or the gas flow being restricted.
Checking the Airflow
The first place to start looking is the intake and outtake of air. The air filter sits just above the carburettor and cleans the air before it enters the mixing chamber in the carburettor. After the gas has burnt up inside the chamber, it vents through the exhaust port and out the muffler. Keeping your air filter clean and replacing it regularly will drastically improve the performance of your engine. While you don't need to clean the muffler and exhaust port as frequently, you should examine them for heavy black carbon deposits that restrict airflow. Newer models also operate with a spark arrester screen inside the muffler. This safety device prevents fires, but it can also be a leading cause in loss of airflow.
Checking the Gas Flow
If your airflow is sufficient, you should check that the gas is being suctioned up into the carburettor and pumped into the chamber. If your chainsaw runs for awhile but quits when it's hot, the chainsaw is likely having gas flow problems. Inspect and clean the gas tank and fuel filter regularly to keep the gas flowing to the carburettor properly. Replace the fuel lines at the beginning of each season. Adjust the low and idle screws on your carburettor to see if you can solve the problem that way before looking further into carburettor problems.
A Dirty Carburetor
Old 029s may have a dirty or worn out carburettor. As the fuel comes in, it mixes with the air, then gets pumped out into the chamber. When the intake or outtake valves get clogged or dirty, the fuel flow stops, and the engine will also stop. Each valve has a screen on the end to prevent impurities from dirtying the carburettor. These screens can be replaced. With your carburettor off, you can pressure test the chamber to ensure there are no air leaks. Installing a carb kit to a cleaned carburettor will help extend the life of an old carburettor.
Practice regular maintenance checks according to the 029's maintenance schedule. Worn or dirty spark plugs can also cause an engine to shut off while running, but more often they will just make the saw hard to start. If you keep the saw cleaned and maintained, you can differentiate minor problems from major problems. Carburettor repairs can be tricky, so if you're in doubt, leave the technical stuff to a professional.