Below, you can find a set of links that will take you to information about medical assistance programs for each state (Medicaid, S-CHIP, or other programs).
Most states, unfortunately, do not offer traditional Medicaid coverage for able-bodied, working-aged adults without children. However, several states do offer other types of medical assistance for working-aged adults, so you should be careful to explore all programs in your state before concluding that you can't qualify for anything.
If you are pregnant or have children, it is far more likely that you will be able to find state coverage. Medicaid eligibility is far less stringent for pregnant women or children under 18, so even if you or a partner don't qualify for benefits on your own, your children (born or unborn) will very likely qualify for coverage if you don't make much money. In fact, graduate student couples in many states have been able to cover their children on state Medicaid programs, because their family income (via graduate stipends) is so low.
So if you're leaving academia with kids and don't have a new job lined up, you should definitely explore whether your kids (at least) may qualify for state coverage.
For each state, I tried to pull a link that provides a gateway to all of the Medicaid eligibility information (and applications) for each state. However, each state offers multiple assistance programs and has complex eligibility requirements for each, so be sure to spend some time looking around your specific site – there may be more information/programs out there than are covered on the linked pages.
District of Columbia: http://dhcf.dc.gov/services
New Hampshire: http://www.dhhs.nh.gov/ombp/medicaid/index.htm
New Jersey: http://www.state.nj.us/humanservices/dmahs/clients/medicaid/
New Mexico: http://www.hsd.state.nm.us/mad/CMedAssisPrograms.html
New York: http://www.health.ny.gov/health_care/index.htm
North Carolina: http://www.ncdhhs.gov/dma/medicaid/
North Dakota: http://www.nd.gov/dhs/services/medicalserv/index.html
Rhode Island: http://www.dhs.ri.gov/DefaultPermissions/HealthCare/tabid/945/Default.aspx
South Carolina: https://www.scdhhs.gov/how-to-apply
South Dakota: http://dss.sd.gov/medicaleligibility/
West Virginia: http://www.dhhr.wv.gov/bms/Pages/EligibleForMedicaid.aspx