It would be smart (assuming a properly insulated and air sealed house--which to your point is uncommon in the US) to build an insulated room with a dedicated hvac zone in the center of the basement or somewhere similar that is basically your "safe room" for such a situation. Would not take much to cool it and has the advantage of being maximally distant from the exterior walls to in case you are worried about radioactive fallout ;)

For most people their car would work fine, of course not everyone does drive. Sometimes cities will open "cooling centers" (school gyms, event venues, etc.) for those cases.