I have not found any specific reference to the age of each of the Pandavas, like Arjuna and Bheema when they fought the epic Mahabharata battle.
It will be also interesting to know the age of Bhishma pitamah during the Mahabharata war as he saw many generations of his own family unfold in front of his own eyes.
Their ages were not mentioned in various TV episodes of Mahabharata.
As per this research article of Mahabharat timeline, Yudhisthira was 91 years old at the time of Mahabharat war. And since all other pandava brothers were younger nearly by one year successively (excluding Sahadeva), there ages at the time of war would have been as below approximately:
Yudhistihra - 91 yrs, Bhima - 90 yrs, Arjuna - 89 yrs, Nakula and Sahadeva - 88 yrs
However, this other site states the age of Yudhistira as 49 years and Devavrata as 140 years. But the first site calculations seem to be more perfect than this one although that doesn't says Bhisma or Devavrata's age. I have found this site while searching today which also says the age of Bhisma was nearly 140 while dying as per his year of birth and death.
Below I have reproduced the timeline found here. Names of years and months can be found on the source of this information -
INTERESTING FACT: Yudhishthir ruled for 72 years in his 127 years of life!
The number in brackets is the time in years from Yudhishthir's birth.
[39] Demon Baka was slain. Princess Draupadi's Swayamvaram took place.
[40] In Panchala Kingdom, Pandavas stayed there for yr. 1-0-15 days
[45] Pandavas were received in Hastinapur. They stayed in Hastinapur for 5 years and 6 months. Indraprastha City was being built during this period.
[46] Yudhisthira was coronated.
[58] Arjuna went on pilgrimage for 12 years. He married Subhadra, and Abhimanyu was born. The Khandava Forest was burnt down. Mayasabha took yrs. 1-2-0 days for construction.
[60] Pandavas entered Mayasabha
[76] Rule in Indraprastha for 16 years. Wrestling started between Bhima and Jarasandh continued for 14 days, and Jarasandh was killed on 14th evening. The Rajasooya Yajna began. The two dice-plays were played in an interval of 4 days.
So Pandavas ruled for a period of yrs. 36-6-20 days
This web page gives a detailed timeline of the Mahabharata, based on the text of the Mahabharata and its associated commentaries. Here is the basic method of calculating the ages of the Pandavas. First of all, as I discuss in this answer, Krishna departed the Earth after he was shot by a hunter who mistook his foot for the mouth of a deer. This event marks the beginning of the Kali Yuga, and it took place in 3102 BC. Now we use the description of the birth of Yudhisthira given in this chapter of the Adi Parva of the Mahabharata:
Then the handsome Kunti was united (in intercourse) with the god of justice in his spiritual form and obtained from him a son devoted to the good of all creatures. And she brought his excellent child, who lived to acquire a great fame, at the eighth Muhurta called Abhijit, of the hour of noon of that very auspicious day of the seventh month (Kartika), viz., the fifth of the lighted fortnight, when the star Jyeshtha in conjunction with the moon was ascendant.
It can be determined using astrological calculation that this took place in 3229 BC, 127 years before the start of the Kali Yuga. So Yudhisthira was 127 years old at the time of Krishna's death. But we know from the Sauptika Parva of the Mahabharata that Krishna died 36 years after the Mahabharata war.
Thus Yudhisthira was 91 years old at the time of the Mahabharata war. And each of the Pandavas were born a year apart. So at the time of Mahabharata war, Yudhisthira was 91 years old, Bhima was 90 years old, Arjuna was 89 years old, and Nakula and Sahadeva were 88 years old.
By the way, as I discuss in this answer, the Matsya Purana says that Krishna was 89 years old at the time of the Mahabharata war, making him younger than Yudhisthira and Bhima, the same age as Arjuna, and older than Nakula and Sahdeva. That is confirmed by the relations of respect described in the Srimad Bhagavatam:
After the Lord bowed down at the feet of Yudhiṣṭhira and Bhīma and firmly embraced Arjuna, He accepted obeisances from the twin brothers, Nakula and Sahadeva.
In any case, using similar astrological calculations, it can be shown that Bhishma was born in 3278 BC, and was thus 140 years old at the time of the Mahabharata war.
By the way, it may seem odd to you that everyone was so old, but you have to keep in mind that people were much longer-lived in the Dwapara Yuga.
If we consider "sacred-texts" as the most reliable source for Mahabharata, then the only ages I could find are those of Guru Drona, who was 85 years old, and Abhimanyu, 16.
Having slain Bharadwaja's son (Drona). Dhrishtadyumna was filled with great joy, and uttered leonine shouts, whirling his sword. Of a dark complexion, with white locks hanging down to his ears, that old man of five and eighty years of age, used, for thy sake only, to career on the field of battle with the activity of a youth of sixteen. [Drona Parva]
A few points to be noted for the pitamaha Bhishma:
He was a young prince, when his half brothers, Chitrangada and Vichitravirya (younger) were born. He was mature enough to take an oath for a lifelong celibacy.
Let's say he was at least 10 years during the oath. At the earliest, Vichitravirya would have been married at the age of 15-18. So Bhishma would be at least 30 years when Dhritarashtra, Pandu, Vidura were born.
It is likely that in earlier times women were getting pregnant at somewhat early ages. They may not have waited to reach the age of 18 years as in the modern era (e.g. Mahatma Gandhi's wife Kasturba got married at 13, but became pregnant only at the age of 17-18).
Based on the above points, we can infer the ages of other characters during the great war:
The curse of Gandhari had wiped out the whole Yadu clan, 36 years after the war. In such sense, Shri Krishna wouldn't have lived beyond 75 years.
Human anatomy hasn't changed much in several thousand years; hence if we hear about over-the-top heights, ages and strengths, then they ought to be ignored either as interpolation or translation errors.
The Mahabharat mentions the time periods that Pandavs spent at different places after they left for Varnavat when they were already adolescents.
The sons of Kunti reached the town of Varanavata and lived there with their mother. And, agreeably to the command of Dhritarashtra, those illustrious slayers of all enemies lived in the palace of lac, while in that town. And they lived in that place for one year, protecting themselves from Purochana.
After they escaped, Bhima married Hidimba and Arjun won the hand of Draupadi in Panchal where they again stayed for a year.
And having obtained Draupadi (as their common wife) they then dwelt there for a year. And after they became known, those chastisers of all enemies went back to Hastinapura.
Then they lived in Khandavprasth for some time:
And the sons of Pritha dwelt there for many years. And they brought, by force of arms, many a prince under their subjection.
Then Arjun went on an exile individually for twelve years:
And Arjuna, that tiger among men, of firm soul, and gifted with every virtue, lived in the woods for eleven years and months.
After this they had the fateful game of dice:
And deceiving Yudhishthira by means of the dice played through the hands of the son of Suvala, Duryodhana sent the Pandavas into the woods for twelve years and one additional year to be passed in concealment, thus making the period full thirteen.
"And the fourteenth year, O monarch, when the Pandavas returned and claimed their property, they did not obtain it. And thereupon war was declared.
Adding these we get somewhere close 35 odd years account of their life after adolescence. So I think their age around 50 years would be more accurate rather than it being around 90 years.
I would like to point out that the classical commentator Nilakantha claims that years mentioned in the Mahabharata do not mean ordinary human years. There is a passage in Mahabharata Virata Parva Section XLIII which says:
And finally Partha, surnamed Swetavahana, hath held it [Gandiva] for five and sixty years.
The translator Sri K. M. Ganguli says in a footnote that 'Nilakantha spends much learning and ingenuity in making out that sixty-five years in this connection means thirty-two years of ordinary human computation'.
