Failed Thermocouple
The thermocouple is a small metal probe that sits in the pilot flame. The heat generates a tiny voltage that tells the gas valve 'pilot is lit, keep gas flowing.' When it fails or oxidizes, the voltage drops, the valve closes, and the pilot goes out.
Light the pilot per the furnace label. Hold the knob down for 60 seconds, then release. If pilot goes out within seconds, thermocouple is failing. Visual check: probe should sit IN the flame, not next to it.
Replace thermocouple. $10-20 part. Most are 18-36 inches long with universal threading. Position the new one so the tip sits inside the pilot flame about 1/2 inch from the burner. Don't overtighten the brass fitting.
Shop Thermocouples