[Occasionally6]

It may be better, if more expensive, to connect the flywheel to a motor/generator and feed the energy in and out electrically, even at the size of storage required in an HPV.

That is how the Williams (F1 initially but now commercialized) and Porsche (which I think is using Williams' technology) work. The flywheel can then be sealed in a vacuum with magnetic bearings, the losses reduced and the flywheel run at higher speed.

I did find on the web (somewhere) reference to a university project where they had built prototypes that were about the size of a large coffee can. It didn't look to be too far beyond DIY either.

There was a Chrysler Sports Car Prototype (Le Mans) a decade or two ago that ran flywheel storage (with an LNG fuelled gas turbine). (It never raced.) That might be worth investigating to see how the gyroscopic effects were handled. The Chrysler Patriot - I think Reynard were involved on the engineering side.

Actually, given the power requirements in an HPV, chemical cell batteries are probably lighter and easier.