The devotion that Jesus called for is radical, but not blind. Note the next few verses: "For which of you, when he wants to build a tower, does not first sit down and calculate the cost to see if he has enough to complete it? Otherwise, when he has laid a foundation and is not able to finish it, all who see it will begin to mock him." Jesus then tells a similar parable about a king preparing for war (Luke 14: 28-32).
Jesus' point is this: we must enter the kingdom open-eyed, recognizing what it may cost. This is much like the parables of the pearl and the buried treasure in Matthew 13:44-46. It is only rational to sell everything one has to buy a property that has a fortune buried in it. In the same way, it makes sense to be willing to pay anything to gain the kingdom.
What about paying the cost of hating your family? In the parallel passage in Matthew 10:37-38, Jesus says "The one who loves father or mother more than me is not worthy of me; and the one who loves son or daughter more than me is not worthy of me." In Matthew's context, it is clear that Jesus is warning about the division that can come in a home when one person becomes a disciple of Jesus and others do not. In many cases, new believers can feel rejected, or even be expelled from their families. In such cases, Jesus says, one must decide which relationship to keep. In Luke 14 and Matthew 10, Jesus is saying that if you have to choose, choose him. Ideally, we don't have to choose - our non-believing family members tolerate our faith, or come to also trust in Jesus. But if they force us to make a choice - choose Jesus, every time.
Both passages remind us that the same is true of life itself. In most cases, the non-believing world allows us to keep living if we become Christians. But if it forces a choice upon us of living without Jesus or dying with Jesus - choose Jesus, every time. Easy to say, not so easy to do.
The picture: Luke 14:26, from The Brick Testament.