A state may not not be a debtor under the U.S. Bankruptcy Code (i.e. "file bankruptcy" as that term is commonly used). A municipality may be a debtor (a Chapter 9 case) only if specifically authorized by state law (See 11 U.S.C. Sec. 109). The City of Bridgeport, CT tried to file a bankruptcy petition about 19 years ago and, at the urging of the State of CT, was thrown out of court.
I think the short answer is that the state simply stops paying its obligations. To the best of my knowledge, during the Great Depression only one state (Arkansas, I think) defaulted on its General Obligation bonds. There have been no state defaults since. And municipalities rarely default. See this Moody's special report
http://www.moodys.com/cust/content/...licy+research/documents/current/102249_rm.pdf which was admittedly published prior to the latest crash, but has some good historical information.