If it truly made the difference between having food, shelter and medical care for my children then I certainly would without hesitation.
However, I cannot imagine a scenario in our modern economy in the US where selling my ring could do anything more than put a stop gap of a month or less on such a dire situation. I think my time would be much better used to find more stable ways to insure for my family's safety (I am not just talking about getting a job, I know they are very hard to find in some areas right now, but also about finding what aid is available to insure that until I can take care of them myself my kids will be fed, sheltered, etc.).