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.
The best way I've found to tame them is to use a basilosaurus,then lure a single electro away from the group and have it shock you (on the mount). Wait about a second for the shock to end and quickly jump off and spam E (keep biotoxin in the 0 slot). *A large school of these can quickly stun and immbobilze larger mounts and knockout players fairly quickly, so use caution!*
Use a basilosaurus. Wear scuba goggles,tank,flippers and then wear riot pants and gloves for torpor reduction. Only try and tame one at a time. Wait until it is 2/3 done discharging, and then hop off and feed it bio toxin before it runs away. When you hop off a basilosaurus, the electric immunity lasts a short while after getting off.
So we tame a 110 plesi then go to get pearls and oil from underwater caves then a group of eels come out of nowhere and stun it. Since we couldn't do anything, it died. Then to our luck that day, a squid comes behind us and we die too and didnt get our stuff back...
Just tamed two on official. I built a small 2x6 box with a Dino gateway out of stone. Park my Dino nearby, then swim with flippers to gain aggro. Kited 4 into box, and i escaped through doorframes. 2 somehow got out through a hole in the ceiling before i could patch it. After a death and some shit, it's was super easy to feed them biotoxin without getting shocked. The stone walls protect you fully. I don't see an easier way if you build your own right.
Build a double sided pen on ocean shelf. Riding an ichy, lure the eel close to the pen. Ride ichy into one side and close door. Once the eel moves into other side, close door trapping it inside. Once it starts poking its head out, feed it till tamed. You may die a few times until you master this technique.