Even on cloudy days, solar panels still work but their output is obviously lower and can drop to as low as 10% in comparison to the current made in the ideal conditions of optimum panel position and cloudless skies.
A feasible plan would be solar panels charging decent quality leisure rated *lead/acid batteries. These can withstand the repeated heavy discharge & recharge cycles, which car batteries cannot and so will quickly fail.
This arrangement can be supplemented with a small diesel generator running on used vegetable oil which is simply processed into 'Biodiesel' or in some engines used 'as is'.
If your electrical energy needs are modest, a Robin DY23 diesel driven generator would be adequate to charge the batteries when sunshine alone is not chucking enough electrons at your solar panels.
Solar panel efficiency is improving (in the mid 70s 1% efficiency was the norm) so by the time you decide to do a Grizzly Adams, panels may well be good enough to meet your requirement without any extras.
By trade, I was a Citroen mechanic, but the idea of self-sufficiency has intrigued me since I was a teenager. If you don't like the idea of relying on free used veg oil from take-aways, a wind-powered generator and a solar panel array would surely be enough to keep the batteries charged for close to all of the time?
Good question.
*Lithium-Ion cells are the modern way to go, but their main advantage is superior energy density, which is good for electric vehicles, but once you've lugged the lead-acid batteries to your island, weight is not a concern.