Obviously if we put live electricity in the sea, the whole ocean wouldn’t then become dangerous with it. At what point does the electricity stop? What stops it?
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You’re kind of coming from the wrong angle. You’re implying that when we put “electricity” into the water, it already goes somewhere, and that there is a point at which it “stops” going somewhere. This is kind of true, but not in the way you think. Everyone else has explained why it’s not true, so I’ll explain why it is true.
For those who understand electric current, ignore all sign convention because it’s confusing and unnecessary for the basics.
To “electrify” a body of water, you have to either put electrons in or take electrons out on a large scale (since that’s basically what electric current *is*, the movement of electrons) which will then spread out and push each other (and other electrons in the water) away from each other. With the extra “oomph” from the current injection (the voltage) there will be a new steady state: either the electrons will find a new place to go (for example the earth or the other terminal of the battery) or, if there is not a place for these electrons to go, then eventually they will bunch up and push back against the source of electrons that’s trying to push them into the water (and a higher voltage source can push more electrons in. This is essentially a capacitor, and the key difference between a capacitor and a battery; a capacitor stores electrical energy and a battery stores chemical energy that it converts to electrical energy.)
The electricity “stops” when the electron injection gets so far away from you that it can’t have a major impact (like sunlight getting dimmer and dimmer the farther away you get in space.)
Things get pretty confusing because we usually say electricity “has” to go back to where it came from to “complete the circuit,” but this isn’t strictly true, and we take advantage of this all the time (it’s called a “floating ground” or a “floating potential” if you’re curious.) The earth is so large that we can treat it as both an infinitely large sink and source for electrons for all conventional purposes, which is why we use it as a global connection for everything (“earth ground.”) However, if you don’t provide a path for the electrons to return to the source, then you aren’t replacing the electrons, and the electron source will eventully have its own force trying to pull the electrons back in, meaning it will get weaker over time as it gets harder to push the electrons out while they get pulled back in.
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