Is it possible to power a very small jet using batteries?
On some scale, of course this works -- look up "EDF", Electric Ducted Fan R/C models of jet planes.  For something the size of a Honda Jet, you still likely wouldn't get the performance (and certainly not the range) that you might desire.  Go smaller, and something in the size range of a BD-5J/Acrojet, which originally flew on a very small engine, might support being converted to an EDF -- though the flight endurance would likely still be very short.

EDF would be suitable for a motor glider or self-launch sailplane, neither of which depend heavily on long power endurance for normal operation (but neither of which are expected to fly hundreds or thousands of miles at hundreds of miles per hour, like a Honda Jet).