Could be worth noting that the pay of an entry level scientist or engineer is less than 1,000$ a month, probably making less sense to someone from outside India. It's because of a low cost of living in India.

Apply at the ISRO website.[0] They have vacancies for all kinds of positions.

[0] http://www.isro.gov.in/careers

Do you have to be born in India?

Yes unless you can provide something Indian citizens cannot provide like expertise in some of the systems ISRO doesn't have proficiency in. I don't say these in demeaning way. India is a 1.26 billion people country and pays less when you compare to salaries of NASA in USD. There's also the aspect of ISRO also being important research organization for defence. It's more like US DRDO and NASA both in a single entity. As a national security risk it's mostly Indian citizens with SMEs on contractual basis

Not true. ISRO is not NASA and "US DRDO" (this should be a clue. DRDO is separate) rolled into one. ISRO has been a civilian program to avoid MTCR issues (although it did not help with GSLV) with DRDO running IGMDP (https://en.wikipedia.org/wiki/Integrated_Guided_Missile_Deve...)

FAIL! As an Overseas Citizen of India, this is extremely disappointing!