Heroine, just a few things to stir into the stew pot.
First off, if your car is unreliable, how do you intend getting back and forth to clients? The worst thing you could do is not show up for a gig.
Second, everybody's idea of a decent income is different. But your numbers, even if feasible, don't add up. Let's say you did, indeed, get five clients, each of which was willing to pay you $50. That's $250/week. Annualized, you're only talking 12 grand a year gross, from which you have to deduct all the costs of doing business (i.e., insurance, transportation, amortization of capital goods, possibly food cost, advertising, licenses, taxes, etc.). Doesn't leave a whole lot in your pocket. And that presumes you will attract that many clients, which isn't as easy as you seem to believe.
You also have the problem of convincing that $50/week potential client that you bring something to the table that he/she can't get just by ordering take-out. With neither formal training nor experience, how do you intend climbing that hurdle?
BTW, I also question this conclusion: I've learned that MOST personal chefs don't have formal culinary training! If by "formal" you mean "classroom," that may or may not be true. But those who don't have classroom training, for the most part, make up for it with a history of in-the-industry experience. That is, they've worked in restaurants, for caterers, or at other food-service establishments, and learned on-the-job. Just guessing, but it wouldn't surprise me to find that the number of successful private chefs who went directly from the home kitchen to cooking for others can be counted on one hand. And maybe have a few fingers left over.
Understand, please, that I'm not trying to discourage you. What I am doing, however, is trying to help open your eyes to the realities. You seem to be letting the dream cloud your vision in that regard.