"The Steam anticheat servers are not responding."

This means something is blocking Valve Anti-cheat (VAC) on your PC. Please refer to this Steam support article for help troubleshooting: Disconnected by VAC: You cannot play on secure servers 

Behind the scenes

This corresponds to k_EAuthSessionResponseVACCheckTimedOut from the ValidateAuthTicketResponse_t callback. According to Steam documentation: "Returned when the players system is not allowing VAC to scan for cheats, and the player should be kicked from the game when this happens."