Is it dangerous to keep the windows closed while someone is cooking with a gas stove?

Unless you are in a new extremely tight house and using every burner on the stove plus the oven, there is no problem. Compared to the amount of air in a room/house what a stove uses to burn is very little and there is always some venting to the outside somewhere.

I have been cooking with a gas oven and cooktop for about 50 years, and have never had a problem with it in the manner asked in the question. You imply that the gas will somehow be floating around in the room where you are cooking, and that you will be breathing it in, or perhaps it will ignite with a spark, somehow? Unless your gas oven and/or gas cooktop are leaking gas into the room, you will be perfectly safe using them in a closed room. During the winter cold weather, I have the room closed, keeping the cold out. That is, until the room gets hot from baking in the oven! Then, I open Windows for a breath of cool air. I have used both the gas cooktop and an electric cooktop, and I love using the gas, and I hate using the electric! When I turn on the eye of the gas cooktop, it comes on immediately. When I turn on the eye of the electric cooktop, it comes on after awhile. It takes several minutes for the electricity to rise to an even heat all around the eye. I'll take gas cooking any day!

Cathy Williams

IN ADDITION:"You can't see or smell carbon monoxide, but at high levels it can kill a person in minutes. Carbon monoxide (CO) is produced whenever any fuel such as gas, oil, kerosene, wood, or charcoal is burned. If appliances that burn fuel are maintained and used properly, the amount of CO produced is usually not hazardous. However, if appliances are not working properly or are used incorrectly, dangerous levels of CO can result. Hundreds of people die accidentally every year from CO poisoning caused by malfunctioning or improperly used fuel-burning appliances. Even more die from CO produced by idling cars. Fetuses, infants, elderly people, and people with anemia or with a history of heart or respiratory disease can be especially susceptible. Be safe. Practice the DO's and DON'Ts of carbon monoxide." epa.gov

http://www.epa.gov/iaq/pubs/coftsht.html