Vectren is here, thinks it smells like a skunk, and so far has not detected natural gas.
There's no skunk smell on the outside of the house...
Edited to add: he says he's detected what seems to be a small leak by the meter, which is in the basement by the furnace. He's going to try to fix it.
Edited again to add: he replaced all the washers and nuts on the meter and says the leak is fixed. He also complemented my sense of smell. The house still smells bad so I left some windows open and went to a local cafe for breakfast.