Is this gray weather suitable for taking two or three-year-old babies out shopping?

It's fine if it's only cloudy, but it's not recommended to take the children out if it's foggy.

Try not to take the baby out even in foggy weather. Don't take your children out in fog or smoke unless you have to. The air quality in that weather is really bad. Even adults with weak children should avoid going out. Don't bury hidden dangers for their health and affect their healthy growth!