Modern Countries existed since Mahabharata times

Apart from India, few other neighbouring countries were mentioned in Mahabharata. Pakistan, Bangladesh, Afghanistan, Iran, Tajakistan, Iraq, Turkmenistan, Kyrgistan, Tibet, Nepal, Myanmar, Sri Lanka had kingdoms with different names.
Present India had multiple kingdoms and janapadas. Almost all of these kings from different countries participated in Kurukshetra war (3139 BCE).
14 Countries during Mahabharata time
1. India (present country) had multiple Janapadas and Kingdoms

2. Pakistan : Sindhu, Keikaya, Madra (Salya, the charioteer of Karna) Kingdoms

3. Afghanistan : Gandhara (Sakuni was from here), Kamboja, Parada, Parasika, Huna, Bahlika, Saka Kingdoms

4. Iran : Aswaka Kingdom

5. Tajakistan : Parama Kamboja, Hara Huna

6. Turkmenistan : Tushara

7. Iraq : Pahlava

8. Kyrgistan : Uttara Madra, Uttara Kuru

9. Tibet : Rishika, East Tushara

10. Nepal : Videha Kingdom

11. Bangladesh : Vanga, Pundra (ruled by Paundraka Vasudeva who disguised as Krishna and was ultimately killed by him)

12. Myanmar : Brahmmdesh (former Burma)

13. Sri Lanka : Sinhala, Trikuta Kingdoms

14. Greece : Yavana Kingdom (Kalayavan died in a trick played by Krishna).
Ionia in greece was called as Yavana or Yona in Sanskrit.
Also Bactria Kings in Afghanistan had origin in Greece.

Few other neighbouring countries like Turkey, Syria may not be having independent kingdoms or recognition as seperate races or kingdoms.

At the time of Mahabharata, Vibhishana was ruling Sri Lanka.
When Yuddhisthira did Aswamedha Yaaga, his youngest brother Sahadeva was sent to south direction till Sri Lanka.
Sahadeva meet Vibhishana and was offered gifts and friendship. Vibhishana accepted Yudhistira as emperor.