What does Mark 3:20 mean?

The "home" referred to here is probably Andrew and Peter's house in Capernaum (Mark 1:29). From the earliest days of Jesus' public ministry, the people know He will eventually turn up there (Mark 1:32–33; 2:1–2).

The Jews understand that God has sovereignty over illness and healing. However, they also believe that ill health is a punishment from God for sin, and that healing is a sign of God's forgiveness. Healing shows that God has favor on the person. It's natural that the people would want healing, not only so their lives will be easier, but as an outward sign that God forgives their sins.

Although the Pharisees believe this, they can't allow themselves to accept that Jesus is the Son of God. They believe God couldn't possibly give authority to a teacher who heals on the Sabbath and doesn't follow the Pharisaical traditions. If Jesus' teachings are wrong, then His healing must be of the same as the pagan sorcerers. This is why they claim His power is from Satan (Mark 3:22–30). God does promise to bless Israel if they obey Him, but that doesn't mean ill health or injury is always a punishment for disobedience (John 9:1–3).

Jesus healing and casting out demons until He is exhausted, or has no time to eat, becomes a common theme in this book (Mark 6:31). There is usually one day a week when Jesus can get rest from this type of ministry—the Sabbath. For as busy as Jesus seems to have been, thus far, on the Sabbath, Jesus has cast out one demon (Mark 1:21–26) and healed one man (Mark 3:1–5) .He's also gotten into a fair amount of arguments with the Pharisees. So, even in Christ, we see a general preference for rest on God's designated day of rest. While Sabbath-keeping is not required of Christians, we would do well to remember that God created rest as well as work.