“at home.” The Greek phrase is en oikos, and it does not mean “in a house” or “in the house,” as if it was Peter’s house. The phrases en oikos and eis oikos (Mark 3:20) are standard Greek phrases or idioms equivalent to our “at home.” Jesus moved to Capernaum after the people of his hometown, Nazareth, tried to kill him (Luke 4:29-31; cp. Matt. 4:13). Jesus either bought or rented a house in Capernaum, because John 2:12 indicates he even moved his family there. Shortly after Jesus moved to Capernaum, it became known as his “own city” (Matt. 9:1).
According to Mark 2:1, Jesus was “at home” when the people crowded his house to such a degree that men had to let a paralyzed man down through the roof (2:4). This was one of the times that the amazing love and compassion that Jesus had for people was clearly visible, because he was much more concerned about the man and amazed by the trust in God those men had than he was concerned about any damage that was done to his house, which was no doubt repaired reasonably quickly.
One of the reasons Jesus likely picked to move to Capernaum was that it was on the Via Maris, the “Road of the Sea,” which was the great trade route from Egypt in the south to Damascus in Syria and on to Mesopotamia (see commentary on Matt. 4:15). The fact that the Via Maris passed by Capernaum helps explain why that city had a centurion living there (Matt. 8:5ff; Luke 7:3ff), which meant having Roman troops stationed in town, and that it had a tollhouse and tax collectors like Matthew so revenue could be collected from the passing caravans (Mark. 2:14). Capernaum was thus a cosmopolitan town with much commerce and opportunity. No wonder, Jesus was so disappointed at the overall reception he got in Capernaum, despite the fact the people were so proud of their town and Jesus referred to it as “exalted.” It was a town with lots of people, yet such a small group of them really believed. Thus he said, “And you, Capernaum, will you be exalted to heaven? You will go down to the grave, for if the mighty works had been done in Sodom that were done in you, it would have remained until this day.”
Jesus Christ chose Capernaum to be his home town after he left Nazareth; he chose a cosmopolitan town where there would be plenty of opportunity to share the Word and reach others, locals as well as people traveling through the town, and also the opportunity for others to more easily reach him.