It is perfectly natural for other family members to be concerned about their loved one(s) taking off somewhere they know little or nothing about with a bunch of strangers. But first, you need to establish what is behind their objections.
If your family members are not Christians then you need to be sensitive in how you handle the situation. But if this is the case then they may never agree with any Christian activity you engage in or understand why you would want to be involved in world mission. Even if your family members are Christians they still might not understand why you are taking this step.
If their concern is for your protection and safety, then do what you can to address their concerns. Get the mission agency you’ll be serving with to talk to them and give them every possible reassurance. Involve your church leaders if necessary. Talking to others who have had similar concerns may also help them.
If at the end of this they still object, then you need to ask yourself questions about what’s most important. If God clearly has placed his call on your life for Christian service - and your church endorses this - then you may need to just go ahead and pray that your family come round to the idea in time. However, God asks us to respect and honour our families, and so you should do what you can to protect your relationship with them whilst explaining why you feel it is right to go ahead.
At the same time, seek advice from people you respect, or others who have gone before you – they may help you find a way through the difficulty.