Ok, let me start by saying that I have absolutely no HVAC training and probably even less electrical training; with that said, I do know how to read and can certainly trace lines on a wiring diagram.
Last night I realized it was about 59 degrees in our house and about 20 degrees outside with the outside temperature falling -- not an ideal situation for winter in Colorado. I held my hand up to a heat register and found that while it was blowing air, it was not blowing warm air, so I turned the thermostat up and went down to look at the furnace and found that the jets that normally are suppsoed to be creating combustion (read "fire") for the heat were absolutely not on and there was no heat around the unit at all. Furthermore the little LED light was blinking, which did not seem like a good thing.
My wife came home and I told her that we needed to call someone and she was actually the one that decided to Google the problem with the model number of the furnace. Turns out that this particular error has to do with some sort of open limit switch or some such nonsense. Essentially what you do is turn off the power to the furnace, find the limit switch on the wiring diagram, and then trace back the wires from the switch, ensuring that each wire is fully seated -- the wiring inside of a furnace uses a bunch of bayonet-type fittings, which means that the female side of the wire needs to be firmly fitted to the male side that is attached to the fan motor, furnace, etc. or you wind up with an open loop. Additionally, the limit switches themselves look like a khaki-colored dime and are located near the jets; the switches have 2 wires going in with the bayonet-type fitting and there's a little "button" (I use the term "button" very loosely because it is really nothing more than a little piece of thin rectangular plastic sticking up from the middle of the wires that maybe depresses about 1 millimeter when you press it) that you push to reset the switch (or furnace has 2 of these switches and there are apparently usually 2 or more switches depending on the furnace).
The first time we fired the furnace up, nothing happened, so I went back and did an absolutely thorough trace of the wires, finding that there was one fitting that was slightly loose -- I removed the wire and fully re-seated it. When I turned the power back on, the jets fired up after about 30 seconds and then the furnace shut itself off; I had forgotten to reset the limit switches after reconnecting the wire. After shutting the power off, resetting the switches, and firing it back up, the furnace stayed on and has been running well ever since.
I can't tell you how happy I am to have heat and to not have spent some ridiculous amount of money for an emergency heating call, and now knowing what was involved in fixing the problem, I would have been very upset in having paid someone to come check wire connections.
I'm not saying that it is appropriate in every situation and I am certainly not saying that there is no danger associated with repairing complex systems in your home, but it may be worth the time to do some searching before you make any calls they next time you have a problem in your own house.