It doesn’t smell like the “natural gas” smell, it smells like gasoline. What could this be and is my house about to explode?
Is this “call the repairman” time or is this “call 911” time?
Have you looked around to see if anything has spilled in your house that the HVAC system is spreading?
Also, if you have an attached garage, check in there.
Lastly, go outside and see if the ambient air smells like gas. In which case it might be coming in through a drafty window or, in some cases your furnace may draw in outside air (by design).
Basically, check the easy stuff first, go see if you can find the source first. It might be nothing to worry about or, at least, something you can rectify yourself.
ETA, I assume (because you would have mentioned it, right) that you don’t, like, live near a gas station?
We had tried all that, and on the advice of a firefighter facebook friend of mine, we went ahead and called them. They used some devices to sniff the air and didn’t find anything that pinged the sensors.
The smell is like paint thinner, now that I’ve had some time to smell it some more…
But the fire department said it’s safe to keep running the heater (it’s definitely coming from the cellar, and is coming up into the house through the heating vents, but they found nothing indicating a safety issue.)
Guess we’ll call the HVAC people tomorrow? Dunno. No idea what could be causing this.
Glad your friend suggested calling the fire dept. That’s what I was going to suggest. The fire department would much rather everyone be safe than sorry.
Do you have an oil (kerosene) furnace? Did you get a delivery today? Could be a gummed-up (I think it’s called a) nipple.
Frylock, did you survive the night?
No we all died and now I’m a scary ghost!
(We’re okay, and the smell is gone.)
Well, that’s good news, but I HATE it when something “fixes” itself, cuz you know it’s going to come back some time when it’s really inconvenient–3AM Monday morning, say. 
Every once in a while I smell what is reminiscent of fuel oil when our natural gas furnace is running. There are still homes here that heat by this substance, so it may be that it’s in the air from a recent fill.
If there was nothing new put in the basement… and there was nothing stored in the basement that has that smell… and you can find nothing in the basement that has leaked…
then it has to be that something from outside the basement spilled & leached through to your place by a wall or a ceiling or the floor or a drain.
Could something in your shed or your garage have spilled?
Any chance that one of your neighbors was painting, got lazy during clean up, and dumped their paint thinner on the ground between your houses?
Does your basement have a floor drain connecting to a sewer where some idiot might have dumped gas or paint thinner?
(Yeah yeah, I know… its gone. Still, finding what it was is a fun parlor game.)
I’m with Burpo. When the oil nozzle clogs, it results in a powerful fuel-oil/rancid smell. If you have an oil furnace, you should get a repairman out to your place ASAP.
Rocketeer, thanks for providing the correct term (nozzle); Guess we can see where my mind was.
After the third time I had the repairman out for the same malady (clogged NOZZLE), he took pity on me and bestowed a piece of knowledge that eliminated future visits from him: when you get a delivery of fuel oil, especially if you have an older drum, wait 3-6 hours for the sediment to settle before using the furnace. Worked like a charm. YMMV.
And yes, that stench is the worst.