These are passive tames. In my experience trying to tame a 145. You will find having a Basilosaurus will provide you with a mount immune to the shock.
Wear a combination of Riot and Scuba gear. Bring multiple sets. They will destroy your tanks and flippers if you get hit enough times. Lone eels do not deal enough torpor to be a substantial threat with the riot gear.
The hard part is feeding them. They will glow bright blue (without goggles on) when they have built up a charge, and charge you. Discharge a 7 second stun, bite and then flee.
Lure them into an enclosed area. Such as a trap for water tames or a river. This will reduce their mobility and make them an easier tame than in the open ocean. Because they will flee the majority of the tame if they have the room to do so.
-Results of 3/20/17 taming test on Xbox official servers.
Don't lead your new tamed eel home while skimming the surface on your "whale"... The eel will surface itself at some point and instamtly die from being out of the water. Took me 3 eels this way before I figured out how they were dying without getting into combat.