How does a firm raise cash when their current short term obligations are greater than their current assets?

A firm's current ratio isn't the only criteron on which a loan can be based... in fact, depending on the nature of the loan, being upside down in the current ratio might be of junior importance.

Loans are also based on, e.g. the value of long-term capital assets; cash flow prospects; guarantees of subsidiaries or shareholders; production contracts with AAA-credit customers;...