Why not? You can set up a family forum, to share stories, pictures, etc. That's household-y and personal. Fits the definition from the law pretty well.
There's some thinking about what constitutes purely personal activity ( https://ec.europa.eu/justice/article-29/documentation/other-... ) usually the test is whether it can potentially reach anyone in the public, is some financial/professional gain for the operator, etc.
> I'm not so sure. It certainly won't be if you have any third party services running on it.
Those are handled in the text too. Basically the controller / provider / operator of said 3rd party service has to be GDPR compliant, not the user. (So if you fire up a WordPress blog, you probably don't have to worry about it.)
As I read it specifically doesn't cover things which are not economic activities and not professional activities. You running a website yourself and not as business may or may not fall into that. It is not necessarily the colloquial definition of personal.
A personal website may have donations, may have ads, may act as advertising for your professional career, may be used to find jobs for yourself, may be used by people to trade items to each other, etc, etc. Those may be covered by GDPR and without a lawyer (ie: money) I have no idea.