What an interesting question. In my case the answer is: not really. My Dad was a dentist (now retired) and my Mum a nurse. Among my grand-parents, two were teachers and one was an engineer (in a textile factory).
So no science but both my parents were very keen on education (they used to make me do lots of homework – or study – every night). That was a bit of a pain but it did give me the habit of working hard.
They also encouraged me to take an interest in the world – I think perhaps this is the best training of all for a young scientist.
I’m the first person in my family ever to go to university to study for a degree. My granddad built the steelwork for buildings so he was an engineer and my dad also worked in an engineering company but there are no other scientists in the family as far as I know.
There’s not a family history of science, but I think both my parents and all four of my grandparents encouraged me and were pleased when I did well in any subject at school. My mum’s parents were both headteachers (and my mum’s dad had an MSc in Psychology, I think). Both my brothers have science PhDs, which is unlikely to be a coincidence, but I’d put it down to the fact that we were all encouraged to think for ourselves when we were growing up. My folks split up when I was very little, and I think my stepdad, who was a science fiction enthusiast and computer operator (in the very early days of computers) had a big influence too.