In some sense a state bankruptcy would be meaningless. A bankruptcy basically cancels some or all of a debtor's debts, subject to various rules and agreements. Although the Contract Clause prevents a state from unilaterally voiding its debts, a state can only be sued in its own courts, so there's no way to compel a state to pay a debt it doesn't want to pay.
On the other hand, a municipality can declare bankruptcy under Chapter 9 of the bankruptcy code, if the state that chartered it allows it to do so. Until the 1930s if a municipality didn't pay its debts, the only recourse was a state suit to force the municipality to raise taxes, which had blood/turnip problems. So Chapter 9 now offers the option of Federal oversight, which has the major advantage (for the municipality) that the Federal government can void at least part of its debts.
So let's imagine, hypothetically, that there's a state with a $25 billion hole in its budget, a short legislative session, a deep reluctance to tax at a level adequate to pay its bills, and a tradition of financial gimmicks. We'll call it Saxet. Chapter 9 says a municipality can file for bankruptcy, which is "a "political subdivision or public agency or instrumentality of a State". The exact definition of an instrumentality is a big vague, so let's say the Saxet legislature passes a law defining the state itself as an instrumentality of the State of Saxet, and authorizing it to file under Chapter 9 any time it wants. Then what? If they show up at the Federal bankruptcy court, does the court have to accept their filing? Who decides?
If you know any constitutional lawyers, feel free to ask them what the answer would be.