Where are they getting the engine? You can get model aircraft jet engines with enough power for this.[1] Various human-carrying craft have been built that way, such as the Zapata Flyboard.[2] But the engines have a lifespan measured in hours and not enough reliability for carrying humans. That's why all the demo videos of these things are over water, or tethered. People have been trying to build small, cheap jet engi…
Another important reason is fuel consumption. Jet engines in general are extremely thirsty, AFAIK the only reason they became (unexpectedly!) competitive with piston engines is that they allow the planes to fly so much higher where the air is thinner.
In addition, small turbines are even less efficient than big ones.