Is it possible to power a very small jet using batteries?
Yes, with some major provisos.
It is misleading to use the word "jet" because a jet engine requires combustion of hydrocarbon fuel to create heat to expand exhaust gasses to drive a compressor and produce direct thrust and/or to power a bypass fan. An electric powered aircraft engine would most likely just use electric power to directly drive a ducted fan.
Photo by Varmin CC BY-SA 3.0
The Airbus E-Fan prototype first flew in 2014 and was the "first all-electric two-engine aircraft" to cross the English channel. I don't know what it's range was but assume it was very short. I think 2 hours was expected for later versions but the project seems to be cancelled.
The motors moving the fans are powered by a series of 250-volt Lithium polymer battery packs
These were described by Airbus as Li-Ion 18650 cells with a total battery weight of 167 kg
These battery packs have an energy density of 207 Wh/kg which is 0.745 MJ/kg
By comparison aviation kerosene has an energy density of 46.36 MJ/kg
Airbus seem to be working on an "E-Fan X" hybrid aircraft
Image from Airbus press-release
The E-Fan X hybrid-electric technology demonstrator is anticipated to fly in 2020 following a comprehensive ground test campaign, provisionally on a BAe 146 flying testbed, with one of the aircraft’s four gas turbine engines replaced by a two megawatt electric motor. Provisions will be made to replace a second gas turbine with an electric motor once system maturity has been proven.
This uses a two megawatt generator to power the ducted fan. So the power source, or more accurately energy reservoir, is hydrocarbon fuel, not batteries.
The purpose of this aircraft is to develop electric propulsion technology, it isn't a prototype passenger transport.