Java constructors are special methods that allow for the initialization of objects and can significantly enhance code organization and cleanliness. However, constructors can throw exceptions, either automatically when invalid data is passed or explicitly for customized handling, to prevent the instantiation of objects with invalid data. Proper exception handling in constructors is crucial to avoid bugs and ensure system resources are not wasted, particularly when an object is partially constructed. Best practices include throwing exceptions before any resource access within the constructor or using mechanisms like try-catch-finally or try-with-resources to release resources if an exception is thrown. The text provides examples of constructors throwing exceptions to ensure data validity, such as checking parameters like age or grade, and suggests using tools like Rollbar for real-time error tracking and management to enhance confidence in deploying production code.