Tom, I think the tanks are usually rotomolded polythene and not much bonds to that. There are some new epoxy sticks (i.e. like plumber's putty) that are targeted for PVC and other plastics but I don't think polythene is on the list, or that the epoxies are rated for drinking water use.
Traditionally you would buy a plastics welding kit, which includes a heat gun and plastic rods. Match the rod to your tank material and you literally weld in new plastic to fill the crack/hole and if you've done it right, you have it repaired.
But the simpler and cheaper route is probably to apply a patch or plug, which is a perfectly good permanent solution. If the hole is small, install a stainless or nylon bolt through it with a rubber or buna o-ring under the head, end of story. If there is a crack, not hole, drill a hole to relieve the stress at each end o fhte crack, the bolt a small stainless plate over it (plate inside the tank) with 4-6 bolts holding the stainless over a piece of gasket material. Again, end of story.
The tank CAN certainly be repaired, you just can't crazy glue it and walk away.
One way to clean a tank that you can't reach inside of, is to use something like Polident and let it soak. Or, pour in coarse rock salt (deicing salt) or coarse koshering salt, and vigorously shake the tank. The salt crystals are just as sharp as an abrasive media--but you can water-wash all of them out afterwards.
"Makes sense: how could this stuff leak, without impact? " Bad casting. The folks who make rotomolded kayaks claim they have an incredibly large defect rate from bad castings, so they have to charge high prices for the ones that are cast properly. (Ahuh.)