Nuclear batteries
The chosen answer is indeed, realistically, the best one. Superconducting solenoid batteries allow for the densest theoretical possible energy storage for electrical power, limited only by the chemical binding force of atoms to prevent it from flying apart from Lorentz forces. With something like aggregate diamond nanorods, you can reach energy densities of 20 MJ/kg easily. And while it doesn't look that impressive compared to, say, gasoline (one of the most energy-dense fuels out there, actually), remember that gasoline + oxygen is not as impressive, that you can extract as fast as you want, you don't need a bulky engine to extract said energy and that said bulky engine would be at best something 50% efficient anyway.
High-temperature high-power superconductors aren't that big of a handwave, nor even room-temperature, in fact, for fare future tech, if you don't want to bother with cryogenic equipment.
Also, it will turn all its stored energy into heat (aka violently explode) if damaged, heated too much or overloaded, which is always good for SF tech. Also also you can actually call it supersolenoid, which makes for nice technobabble.
But that's not enough! (After all, it never is.) We are talking about far-future tech! We don't want to be limited by weak atom bonds!
If you are OK with a bigger handwave and wants moar power! in storage, you can go for nuclear isomers. A nuclear isomer is an otherwise stable atom nucleus that is in an excited state. At some point it will decay, but unlike things like beta decay, it will only emit a gamma ray - and the process is (theoretically) reversible. And an excited nuclear isomer contains a lot of energy. Like more than a million MJ/kg. Simply put your far-future gammavoltaic cells to turn those gamma rays into electricity and you're fine.
The problem is, isomers come in two categories: the "barely existing" ones, that will revert in a nanosecond, and the "nearly stable" with an impractically long half-life. Stabilizing the former is probably impossible - but wouldn't it be nice if we could induce the latter to revert and emit that sweet, sweet high-energy photon?
Some guys pretended they had managed it with Hafnium isomers by spraying them with X-rays. Alas, this has been discredited since. Hafnium batteries have since gone the way of the Dean drive, the water engine and the EMDrive. Sigh.
But wait, not all hope is lost! Unlike those, the Hafnium battery wasn't dismissed out of hand by anyone with a modicum of knowledge in physics, because it could have worked! Which means that there may be other methods, beyond current technology, that actually work. Maybe with some exotic particle that require a brand new type of particle accelerator.
The point is, while today Hafnium batteries are bunk, they are still believable as far-future tech.
There is also the question or producing those Hafnium isomers, but if you can induce de-excitation, you should know how to induce excitation as well.
Hafnium isomers have 31 years of half-life, which is fine for short-term energy storage. If you need much longer-term, you can use Tantalum isomers instead. With 40 000 MJ/kg instead of a million, they are not nearly as dense (though still much better than supersolenoid batteries), but their half-life is much, much longer than the age of the Universe.
Those tables may be useful for comparison of storage densities, which is on of the main criteria here.
