When you and I were little, being sat in front of a screen meant watching TV, but today there are a whole variety of different screen based activities competing for attention. Common wisdom was to manage the amount of television that young children were exposed to. Whatever the agreed limit, impose it regularly and don't oversexpose your little ones to too much.
Today there are likely to be screens all over the house, in your bags and pockets, possibly in the car too, whether it be televisions, computers, mobile phones, tablets, laptops, games consoles or even hi-fi's and other entertainment systems. Our children are undoubtedly growing up in a very different environment to the one that we grew up in, and it's important that they gain the right exposure to the right devices in the right ways. Children will use computers at school from an early age, so early exposure to them is not necessarily a bad thing. They may soon be using tablets at school, so again, controlled access is probably a better policy than prohibition.
As much as you may dislike technology, or feel uncomfortable with the level of exposure in everyday life, you could place your children at a disadvantage in life if you prohibit them from accessing televisions, computers and even games consoles. As with television in days gone by, adopt a screen policy where you allow 'screen time' at certain times during the week. Make this time optional, that your little ones may sit in front of a screen if they wish, but they may choose to play or do something else instead, don't mandate that they must sit in front of the telly, or play educational games on your phone.
Whether you grant an hour or two each day, half an hour a week, or somewhere in between, the overall time isn't so important. More important is that you manage expectation, enforce the limits and try to make this time constructive. Perhaps you might choose to watch a couple of regular programmes, or to play certain games on a television or computer. Try to steer your little ones' attention towards educational content and remember that even games can be construed as assisting development if they help develop coordination or understanding.
If you think our children have a shockingly different life to the way that we grew up, being exposed to electrical gizmos and gadgets at every turn, just spare a thought for the world that your grandchildren will inhabit!