Can overstayed J1 Visa get a green card through marriage to a US citizen?

The fact that your U.S. citizen husband now is unemployed will not stand in the way of eligibility for a marriage-based adjustment of status case by which you would become a Lawful Permanent Resident (would get a "Green Card"). If the petitioner (for example, your husband) has insufficient documentable income, then it may be necessary to have a joint sponsor in the application process. Generally any U.S. citizen or Permanent Resident who has sufficient documentable income and who is willing to assume the responsibilities contemplated in the Affidavit of Support can serve as a joint sponsor. If the J-1 visa you previously had was subject to a two-year home residency requirement, then note that you either will need to comply with that requirement of get it waived in order to be eligible for adjustment of status. If the J-1 visa did not involve governmental financing/support, then achieving a waiver may not be too complicated. Of course, there are many other details that determine eligibility. It would be wise for you and your husband to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.