Rome and Han
At the two ends of Eurasia, two states grew into vast empires at almost the same time.
China was united by Qin in 221 BC and held together by the Han. Rome beat Carthage, swallowed the Hellenistic kingdoms and, under Augustus, became an empire.
By AD 1 they ruled roughly half of humanity between them. They knew of each other only faintly, through the silk that travelled the roads in between.