This is the American Airlines small dog carry-on bag, available from Petco.
Slightly different versions, made by the same company (Sherpa), are tagged with other airline monikers, such as Delta, but I think they are all interchangeable.
As you can see, a small Jack Russell terrier can fit inside these carry-on bags, provided the dog weighs less than 16 pounds and is not too much over 11.5 inches tall.
Have dog toy, will travel.
I have several friends out West who bring a dog East with them to hunt every year, and this is how their dogs come. Seems to work! I have dug with their dogs up to 2 pm in the afternoon, washed the dirt out of the dog's eyes, and sent them off on a 4 pm flight with a little dirt still stuck to the both of them. And no, I have no idea what we would have done if the dog had gotten skunked!