Making electrons flow is what creates electricity. So if we are able to put electrons into a system, close the loop so the electrons have no where to go, then the electrons would be constantly bouncing around together. This would cause charge to flow around and create electricity.
You're getting lot's of things wrong here.
- The term "electricity" is extremely vague and not technical at all. There is no physical phenomenon called "electricity" nor any physical quantity called like that.
That term is a relic from the past when people didn't know much about current, voltage and other macroscopic electrical quantities. Let alone electromagnetic theory. That term can be used correctly only in non-technical/scientific contexts or only in the vaguest of sense. In other words: you don't measure "electricity", you measure a bunch of well-defined physical quantities related to electrical phenomena.
From a technical point of view, what you call "electricity" in the sense of "some kind of energy that allows us to do stuff" is called "electrical energy" or "electrical power".
- Electrons moving around don't necessarily generate electrical power. Even assuming electrons were classical particles (and not some elementary quantum particles with all kinds of odd, non-intuitive behavior), a flow of electrons would constitute an electrical current (or a current density distribution, if the body wasn't a thread-like body).
Currents can generate power only if they flow through a system that is able to draw energy from them. The electrons have to lose energy to provide power to the "outside world". In other words, they must travel across some sort of potential barrier. In technical terms, if the electrons moving from point A to point B are to generate power, there must be a voltage across those two points (point A must be at a lower voltage than point B, in particular).
- For electrons to "bounce around" without having "nowhere to go", you must put them in a body that is both an electrical conductor and that is insulated from the rest of the world.
(If the body weren't a conductor, electrons wouldn't be able to "move around", they would stay "locked" to the atoms of the body).
In this scenario electrons, even if moving, wouldn't be able to generate power, because any point of a perfect conductor is at the same potential. So there is no voltage drop, hence no power.
Of course I have neglected lots of additional problems that arise when you consider the true nature of electrons, which are not tiny weeny balls of matter, but quantum particles to which classical reasoning cannot be applied coherently.
BTW, it has already been hinted to by others in this thread that you can have a continuous flow of current in a ring-shaped superconductor body. The electrical resistance is virtually 0 and the electrons don't lose energy while moving (ideally). There have been experiments in which such a "circular current" has been maintained for days (IIRC) without further external energy provision.
- Conservation of energy. Moving electrons (i.e. electrical currents) don't generate power (energy) out of nowhere. They are just a convenient carrier for power (energy) generated elsewhere. There are machines called electrical generators (batteries, alternators, photovoltaic generators, etc.) that take energy from another source (chemical, mechanical, electromagnetic, etc.) and turn that energy into electrical energy by putting electrons in motions. Then electrons are just, as I said, convenient carriers, since they can easily travel in wires to power up electrical loads placed even kilometers away from the source of power.
Remember: energy in nature cannot be created. It can only be transformed from a form into another. This is called the Energy Conservation Law and it's a basic principle of the Universe that no validated scientific experiment has ever been able to disprove.
Any time anyone has "witnessed" phenomena that violated that conservation principle, it has been established by scientific analysis that the phenomena were observed without taking correct measurement into account or that some other experimental mistake has been made (assuming there wasn't bad faith and the phenomena were not part of a scam).
Truly scientifically proving that the Energy Conservation Law is not universally valid would be a discovery worth a couple of Nobel prizes!