Because the island is incredibly rural and even rental cars can have holes in the floorboards (in a charming way), there's just not enough infrastructure on Eleuthera to rent your own boat and traipse around (the blog post above sailed to the island from FL I believe). And being 100 miles long and a mile wide means there are scores and scores of cliff lines to investigate as well.
If you're interested, check out Coco Loba Tours' Facebook page and contact Manex. He's helpful, professional, and if you're lucky, while you're climbing, he'll spear some lobster below you and that's what you can have for dinner that evening.
He also has a website in development calledmycocoloba.com if you're interested. As someone who has used Manex for several deep water soloing excursions on Eleuthera, I can honestly say it wouldn't have happened without him. And to cap it off, he's not a climber, so he'll tell you how amazing you are as he watches you climb the 5.7!