How “safe” is life overseas for single women or families?

Depending on the specific country, life overseas is often much safer than back in your home country.  But this is not always true.

Crime

Serious crime, in the countries where the author has lived, seems to be much less of a problem than "back home".  Be careful until you know for sure, but a good starter rule is, "If you wouldn't do it back home, don't do it in your new country." 

Part of the safety issue is that foreigners often don't know the signals of a bad neighbourhood in the new country.  The author has lived in neighbourhoods in which, "back home", he would have been afraid to go outside after dark, but in the country where he was living - no problems at all.  Very safe. 

BUT, I have to qualify that most of my experience as been in Asia.  South and Central America - and Africa - are different.  Europe - I don't know about.  Korea, Japan, Taiwan, and Thailand - seem to me to be much safer than my experience back in the States.

Traffic

Traffic accidents, and the lack of First Aid skills by local populations, probably present a greater threat to your physical well-being than crime.

Africa

When I lived in Africa, it was clear to me that crime against people and property was a much greater threat than back home.  Investigate carefully when considering a job there.  I know that is a gross generalization and there are 50 some countries in Africa - but still, I want to say, "Be careful."  

Families and Women

Be as careful as you would back home - at least until you know the areas where you live and work.  But, again, my opinion would be that in most parts of Asia - you are safer here than at home.

Much of the violent crime that does exist in Asia often seems to be business and/or gang-related.  As long as you stay out of those circles you will probably be just fine.