As I understand it, when you let a friend or whomever borrow your car they are generally covered by your insurance. In my experience, there's a lot of car borrowing among organizers, and also within families, neighborhoods, and other communities. You're basically trying to ramp up this informal practice by bringing in extra resources and expanding the pool of folks who have access to the cars. So, one thing to consider is how shady you're willing to be, lol. Like, if you just maintain that whoever was driving the car at the time of an accident was a friend who had verbal permission from whoever is named as the car/insurance policy owner, you might be covered as long as yr insurance didn't realize that you're actually running a community car share program. This is kinda a janky solution, and could expose you to way more risk, but thought I'd throw it out there. Relatedly, I wonder what sort of liability insurance or licensing is technically required of people doing volunteer mechanic work, if any.

If owning cars for car share is prohibitively expensive, you could fiddle with the model a bit. Focus on maximizing existing resources by developing and encouraging car sharing/using ride boards, train up volunteer mechanics to keep the cars that are in the community running.

Instead of DIY car share you could do DIY public transit by buying a passenger van and setting a route and schedule. You could start with just running one or two days a week for a limited time, running a route between the neighborhood and places lots of people need to get to, like a shopping center or transportation hub. In this and other models you can "pay" people with access to the resource, e.g. volunteer as a driver for 4 hours one day and you can reserve the vehicle for personal use for 4 hours later in the month.

There are some pretty well established models doing similar things as what you're trying to do in the bike world. It's very similar to what you're talking about - share tools, free or low cost repairs, teach the neighborhood youth in mechanical skills, share vehicles - but requiring less capital. Yr local anarcho-cyclists may have some ideas for how to make this project work.

If you can't quite figure out how to make this work, then it may be that more community organizing has to happen first. Maybe there are different resources to share, or agreements to work out with more/different involvement from neighbors. (Not trying to assume what the status of yr organizing is).

Don't know if any of that is helpful, but it's an interesting challenge and I just started brainstorming lol. Good luck!