The decline of social connectedness caused by distance and physical limitations severely affects older adults' well-being and mental health. While virtual reality (VR) is promising for older adults to socialize remotely, existing social VR designs primarily focus on verbal communication (e.g., reminiscent, chat). Actively engaging in shared activities is also an important aspect of social connection. We designed RemoteChess, which constructs a social community and a culturally relevant activity (i.e., Chinese chess) for older adults to play while engaging in social interaction. We conducted a user study with groups of older adults interacting with each other through RemoteChess. Our findings indicate that RemoteChess enhanced participants' social connectedness by offering familiar environments, culturally relevant social catalysts, and asymmetric interactions. We further discussed design guidelines for designing culturally relevant social activities in VR to promote social connectedness for older adults.