What does Matthew 10:36 mean?

It's hard to imagine a more terrible scenario than families turning on each other. To share a living space with one's worst enemy creates the worst kind of strife. It cannot go on. If it continues on a wide scale, the society will break down.

What could possibly cause this kind of separation within Israel's families? Jesus says it is Him. His arrival, and the challenge of truth, will provoke exactly this kind of friction within and between the families of Israel (Matthew 10:34–35). Those families will divide over the issue of whether Jesus is the Christ, the Messiah, the Son of God. Those who reject such belief as blasphemy will turn on those of their own household who acknowledge Jesus as the Lord.

Jesus is warning His apostles that this will happen within their own families and those of the people they preach the truth to. After all, Jesus' claim to be the Messiah caused division in His own hometown (Matthew 13:53–58) and with His immediate family members (Mark 3:21). He makes clear in the following verses that if it becomes a choice between family and faith in Him, His followers must choose Him or be found unworthy of Him.