I am baffled by some of the responses, just wow. And I am so sorry you are in this situation. What your husband did is really unfair. Regardless of the impact of depression on his actions, it is unfair.
Of course couples need to both be willing to compromise and support one another's needs. He is not asking you to have a discussion to work out what is best for all involved, he launched you into this position. You do not seem like some selfish housewife who refuses to work. It would be one thing if he said "hey I really need to reduce my work life and change careers, could we talk about what this might look like?" He QUIT HIS JOB. He's leaving you little to no choice.
My husband has a super stressful job, and is becoming more and more unhappy with it. I have told him that if he feels like he needs to quit and/or change careers, I am totally willing to go back to work. I would also be willing to sell our house and downsize to reduce the financial pressure on him/us. Whatever he needs. I don't think he will take me up on this, he sees this as a burden to me. I think closer to retirement age, he may consider downsizing, and I will probably go back to work just to add some extra comfort. And years ago, he started out as a stock broker, and it completely blew up and he was let go. He realized he hated it. He needed a career change. But he refused to sit and figure things out- he got a job waiting tables to bring in some money and found a new career.
So I'm saying that I think you are awesome for your willingness to uproot your lives and support him through this time. But he can't just dump everything in your lap and expect you to take it- this is not the work/life balance expected in families.
However, what's done is done. If he quit- he quit. So what do you do now? Start by telling him you see that he is having a hard time, and you are willing to come together to see this through. You are willing to work- evenings or whatever. Maybe look into working in the school? This way you'll be near your kids and have similar schedules? Tell him you will explore working, but you'd like his support in understanding that being around the four kids is important to you.
Tell him you are happy to help him explore a new direction, but he needs to bring in some income in the meantime. He can work retail or anything right now while he figures out his next move. I really don't get men or people who do nothing to support their families while they "figure things out." So fine, he isn't ready for a new career- great. But he best get ready for some kind of job right now. There are so many resources. You mention he has his degree- are you still in the same town as his college? Most colleges have alumni career fairs or services, have him look into that. Even if you've since moved, I'm sure there is online help through the school. This will provide some direction and a start.
Great that your lease is up soon so you can find something cheaper if necessary. But seriously, get some income rolling or I'm afraid the other poster was right- you are on your way to living in your car.
As a side story, my sister and her husband both worked in advertising. They both got laid off. He announced that he hated 'being chained to a desk' and wasn't going to look for anything else, he would stay home with their daughter. So my sister had no choice but to find another agency. She says she is fine with this, but she often cries. Many nights she Skypes goodnight to her daughter because she is working so late and calls me so sad about that. During a blizzard in NY she had to trudge to the subway because hubby was afraid the car might get stuck and didn't want to take it out. She continues to accept his increasing demands on her. She tries so hard to enjoy being a career woman, and her work ethic is stellar. But my biggest fear is that their daughter will be grown and she is going to look back and be bitter as hell at what she missed out on with her. I don't want that for her, and I don't want that for you. That doesn't mean you shouldn't return to work in some form, it just means that it had better be because you decided it as a family and weren't forced into that decision. Good luck to you.