was or were? "The family" is single, thus you should use "was".
If there are several families, then you can use "were". May be you are confused because the word has a plural meaning, but it doesnt matter. We treat the word "army" as singular :)
I would prefer using "having lunch" instead of eating lunch: "the family was having lunch"
Edit: It's true that sometimes "family" can be used in plural form. There are some words like this, which can be used in both plural and singular form, depending on the meaning. However in this case, since the word is a group of people united under the same roof, we should use singular form.
If you treat "the group" as a unit, it will be singular, if you treat it as a number of individuals, it should be plural.