Can a state declare bankruptcy?

No, there is no authorization for states to do so under the United States Code.

Municipalities are explicitly authorized to declare bankruptcy 11 U.S.C. 109:

"(C) An entity may be a debtor under chapter 9 of this title if and only if such entity-
(1) is a municipality;
(2) is specifically authorized, in its capacity as a municipality or by name, to be a debtor under such chapter by State law, or by a governmental officer or organization empowered by State law to authorize such entity to be a debtor under such chapter;
(3) is insolvent;
(4) desires to effect a plan to adjust such debts; and
(5) (A) has obtained the agreement of creditors holding at least a majority in amount of the claims of each class that such entity intends to impair under a plan in a case under such chapter;
(B) has negotiated in good faith with creditors and has failed to obtain the agreement of creditors holding at least a majority in amount of the claims of each class that such entity intends to impair under a plan in a case under such chapter;
(C) is unable to negotiate with creditors because such negotiation is impracticable; or
(D) reasonably believes that a creditor may attempt to obtain a transfer that is avoidable under section 547 of this title."

11 U.S.C. 101(40), in turn, defines municipality as a "political subdivision or public agency or instrumentality of a State."

Given that municipalities are explicitly authorized to declare bankruptcy and states aren't, a bankruptcy judge would most likely conclude that Congress intended not to make bankruptcy relief available to states.