There is an old dude in the desert near Wellspring, who tells you not to go down nto the sewer right next to him. Naturally, that's where I wanted to go.... but when I try and open the sewer, it says "you do not have access or have not purchased this downloadable content". I have scoured the net and cannot find out how to get this - can anyone enlighten me on how to purchase this extra content? I feel as if I am missing a large chunk of the game....

It came with the Anarchy Edition. There are several sewers to explore. As far as I know there is no other way to get access to them.

There are 9 sewers total. 7 in the first Wasteland, and 2 in the Eastern Wasteland.
They came included with brand new copies of the "Anarchy Edition" when the game was released. But now, I don't know how you'd go about getting them.

Honestly, you're not missing much. They are very short, corridor romps where you fight some mutants and get a few extra items. Nothing huge.