How do networks of relationships evolve over time? We analyse a dataset tracking the social interactions of 900 individuals over four years. Despite continuous shifts in individual relationships, the macroscopic structural properties of the network remain stable, fluctuating within predictable bounds. We connect this stability to the concept of equilibrium in statistical physics. Specifically, we demonstrate that the probabilities governing network dynamics are stationary over time, and key features like degree, edge, and triangle abundances align with theoretical predictions from equilibrium dynamics. Moreover, the dynamics satisfies the detailed balance condition. Remarkably, equilibrium persists despite constant turnover as people join, leave, and change connections. This suggests that equilibrium arises not from specific individuals but from the balancing act of human needs, cognitive limits, and social pressures. Practically, this equilibrium simplifies data collection, supports methods relying on single network snapshots (like Exponential Random Graph Models), and aids in designing interventions for social challenges. Theoretically, it offers new insights into collective human behaviour, revealing how emergent properties of complex social systems can be captured by simple mathematical models.