If my J1 visa with no 2 year rule got terminated and I continue living in the US, would I be able to get social number?

A foreign national may remain in the U.S. lawfully only with a valid nonimmigrant visa. If your J-1 visa expired or otherwise was terminated, then you would be regarded as "out of status," and subject to being placed in removal/deportation proceedings. Note that depending upon the length of unlawful presence, a foreign national may become subject to a 3-year or even a 10-year bar to re-entering the U.S. When a foreign national who entered the U.S. lawfully and with inspection (such as with a J-1 visa) becomes married to a U.S. citizen, the couple may succeed with a marriage-based adjustment of status application process, so that the foreign national will become a Lawful Permanent Resident (will get a "Green Card"). This is true regardless of whether the foreign national spouse may have overstayed a visa, and even if the foreign national spouse may have worked without authorization. It would be wise for you and your fianc?/fianc?e to consult with an immigration attorney who, after learning all of the relevant information, could advise you about immigration eligibilities, options and strategies (this may include supplying documentary evidence, beyond what may be shown on the visa or Form 2019, that there was no two-year home residency requirement associated with the J-1 visa).