There is no explicit age given. But Barristan Selmy's guess seems to be a good one.
There is no reason to believe that oldest MUST mean it is also the biggest. Not all the dragons would have died at the same time, some may have died 2900 years ago. But the longer a dragon lives, the larger they get.1 (All quotes will be compiled at the end).
What we do know?
- Balerion flew West with Aenar Targaryen before the Doom of Valyria2,
- Balerion was the largest skull in the throne room, followed by Meraxes and Vhagar3,
- Meraxes' and Vhagar's skulls were no more than ~415 years old at the time of A Game of Thrones4,
- Meraxes was no more than 125 years old at the time of its death and yet was the second largest skull5,
- Balerion died in 94 AC and was at least 220 years old at the time (because he arrived with Aenar)6,
- Vhagar died in 130 AC and was 182 at the time of its death. It was said to be nearing Balerion's size7,
- Although the second largest skull in the throne room, Meraxes died at least 60 years before Vhagar8
With all the above information, we can make a reasonable claim that Dragons did not live forever, and that if Vhagar was reaching Balerion's size after 182 years Balerion couldn't be 3000 years old, otherwise he would've been FAR bigger. Balerion is also noted to have been the oldest dragon, meaning dragons were not immortal and died after ~200 years of life. This is given further evidence, since Meraxes, at most 125 years old, was larger than Vhagar and therefore the three skulls must've been similar in size, and within at least a few decades, at most maybe a couple of centuries in age. (Although accounts seem to differ whether Vhagar was larger or smaller than Meraxes 3, 6)
Therefore Barristan's prediction was likely a good guess for Balerion's age, although not quite exactly right, it was a close bound. Balerion was definitely not 3000 years old and this suggests that the Targaryens had in fact brought Dragon skulls with them before the Doom.
1 A dragon never stops growing, Your Grace, so long as he has food and freedom.
A Storm of Swords - Daenerys I
2 Of the five dragons who had flown with Aenar the Exile from Valyria, only one survived to Aegon's day: the great beast called Balerion, the Black Dread.
The World of Ice and Fire - The Reign of the Dragons: The Conquest
3 You could have ridden a horse down Vhaghar's gullet, although you would not have ridden it out again. Meraxes was even bigger. And the greatest of them, Balerion, the Black Dread, could have swallowed an aurochs whole, or even one of the hairy mammoths said to roam the cold wastes beyond the Port of Ibben.
A Game of Thrones Tyrion II
4 The remaining two dragons—Vhagar and Meraxes—were younger, hatched on Dragonstone itself.
The World of Ice and Fire - The Reign of the Dragons: The Conquest
As shown below, Aenar left for dragonstone in 114BC
5 Earliest possible birth: Twelve years before the Doom of Valyria (114 BC), Aenar Targaryen sold his holdings in the Freehold and the Lands of the Long Summer and moved with all his wives, wealth, slaves, dragons, siblings, kin, and children to Dragonstone, a bleak island citadel beneath a smoking mountain in the narrow sea.
Death: Matters escalated, and more Dornish seats fell to dragonfire in 9 AC... A bolt from a scorpion pierced the eye of Meraxes, and the great dragon and the queen who rode upon it fell from the sky.
6 When the Black Dread died (of old age, not in war), he did not take a second dragon.
The Rogues are Coming (March 12, 2014), Reply to question (March 12, 2014)
...the oldest and largest of the Targaryen dragons since the passing of the Black Dread in 94 AC.
The Rogue Prince
7 ...she had grown nigh as large as the Black Dread of old.
The Princess and the Queen
8 It was upon the twenty-second day of the fifth moon of the year 130 AC when the dragons danced and died above the Gods Eye. Daemon Targaryen was nine-and-forty at his death; Prince Aemond had only turned twenty. Vhagar, the greatest of the Targaryen dragons since the passing of Balerion the Black Dread, had counted one hundred eighty-one years upon the earth.
The Princess and the Queen