1. The doctor told the boy’s parents that he was no longer in danger of dying—he had made it through the night and his high fever had returned to normal. He was out of the woods.
2. The accountant was trying very hard to find a bank that would loan the company money to stay in business. When he found a bank, the company president was relieved, but the accountant told him the company was not out of the woods yet. 67

The expression is often used to describe no longer being in danger of dying (opt.1). It can also be used to describe situations equally critical, such as the demise of a company (opt.2) or the death of someone’s political life (opt.3).