Work on your business part-time, or put in some extra hours at work to increase your rate of saving for the eventual day when you can quit. It is possible to build a business part-time. It's harder, the odds are stacked even higher against you, and you get a lot less done. But you get to keep your family fed and your health/retirement benefits in place and working for you.
pg covered this in his "Why not start a startup" talk. Every reason except "you have a family to support" he debunks. This is the one that can't be debunked, because starting a business is not guaranteed. It could take months or even years to start seeing revenue, if you ever see revenue. A few grand from YC only makes a few months difference, tops, in the "when do I run out of money" equation. If you aren't already quite confident you have the money in the bank to last until you can make your business pay or at least be sure it's not going to and you can feel good about going to work at a regular job, you aren't in a position to take that risk.
And has anyone considered the cost of maintaining your ability once you've settled into the corporate world? Go for it while at the top of your game, if possible.
The advice was still to work on a startup, but given the situation, to use a different strategy.
Your family's welfare has to come first- Not only is there the loss of income to think about, but health insurance.
Instead, I'd suggest spending the next 6 months living below your means, and putting the money aside. You don't need much, but enough to comfortably live on for a few months, supporting your family.
Price out private health insurance- This is cheaper in regulated states such as Massachusetts, but your situation and location will vary.
After you have a few months salary put aside, then you'll be in a stronger position to take time away from a job to start a company.