In order to be able to start searching for the Nuclear Waste barrels, you will have to first complete the Merryweather Heist and get the submarine. Then, you have to go deep underwater and explore the coasts of Los Santos to find all the Nuclear Waste barrels. There are 30 barrels to be found and you get $32,000 for each found barrel, plus a hefty bonus of $250,000 if you find them all.
To make things a lot easier for you, we have a video below showing you the locations of the Nuclear Waste barrels:
Cheers PowerPyx for creating this amazing locations guide!