By far the most likely cause if you have a standing pilot (pilot on all the time) - is the thermocouple has died. It is a little probe that sticks into the pilot flame, and has a copper tube (or sometimes wire) back to the gas controller - if the thermocouple does not sense the heat of the pilot it prevents the gas controller from turning on the burners. A dead thermocouple reads as a pilot light blown out but will not always cause the pilot light to go out, though it should, but does not always, particularly on older models. Typically about $75-125 labor to replace plus about $10 for the part. Could be dead due to corrosion, internal flaw, or got kinked so it does not work.
Could also be thermocouple is not properly in the flame (about the tip half or 3/4 inch should be in the blue part of the pilot flame), or rust may have fallen on the pilot and be blocking the flame from heating the pilot enough.
Another possibility is a bad gas controller, bad thermostat (either furnace thermostat or household), or a problem in the electrical controller like a stuck relay, or a backup hydrostat (safety thermostat clamped onto the outgoing hot water pipe, if huydronic system) that is telling the furnace it is too hot, whether or not that is true.