The Sack of King's Landing takes place in 283 AC, taken from the Appendix in A Dance with Dragons:
In the 283rd year after Aegon's Conquest, Robert of House Baratheon, Lord of Storm's End, overthrew the Mad King, Aerys II Targaryen, to win the Iron Throne. His claim to the crown derived from his grandmother, a daughter of King Aegon V Targaryen, though Robert preferred to say his warhammer was his claim.
A Dance with Dragons, Appendix, "House Baratheon"
The pregnant Queen Rhaella and Prince Viserys are sent to Dragonstone not long before the Sack of King's Landing when Aerys II heard of Rhaegar's death at the Battle of the Trident.
"Rhaegar met Robert on the Trident, and you know what happened there. When the word reached court, Aerys packed the queen off to Dragonstone with Prince Viserys. Princess Elia would have gone as well, but he forbade it. Somehow he had gotten it in his head that Prince Lewyn must have betrayed Rhaegar on the Trident, but he thought he could keep Dorne loyal so long as he kept Elia and Aegon by his side.
A Storm of Swords, "Jaime V"
Daenerys is born nine moons into the flight to Dragonstone, most likely in 284 AC now, I expand on that a bit more in this answer.
She had been born on Dragonstone nine moons after their flight, while a raging summer storm threatened to rip the island fastness apart. They said that storm was terrible. The Targaryen fleet was smashed while it lay at anchor, and huge stone blocks were ripped from the parapets and sent hurtling into the wild waters of the narrow sea. Her mother had died birthing her, and for that her brother Viserys had never forgiven her.
A Game of Thrones, "Daenerys I"
And there are multiple mentions of fourteen years since Robert took the Iron Throne and Dany turning fourteen.
They were on the far side of the Dothraki sea when Jhiqui brushed the soft swell of Dany's stomach with her fingers and said, "Khaleesi, you are with child."
"I know," Dany told her.
It was her fourteenth name day.
A Game of Thrones, "Daenerys III"
So the series starts fourteen-ish years after Robert's Rebellion which took place from 282 AC to 283 AC with Dany being born not long after in 284 AC. Therefore, it takes place in 298 AC.