As one delves into the intricacies of veterinary science, one can't help but be intrigued by the complex yet crucial role played by Emergency Veterinary Hospitals. These institutions, in essence, can be likened to the emergency departments of human hospitals, however, they cater to the immediate and sometimes life-threatening needs of our non-human companions. In order to understand their operation, it is pivotal to comprehend their structure, the array of services they provide, the challenges they face, and the significant impact they have on the broader landscape of animal health care.

The structural anatomy of an Emergency Veterinary Hospital is a blend of design functionality, cutting-edge technology and the art of compassionate care. Typically, these facilities include a reception area, examination and treatment rooms, surgical suites, diagnostic labs, and hospitalization wards. The size and complexity of these hospitals vary widely, contingent upon factors such as geographical location, clientele demographic, and specializations offered.

The services rendered by these hospitals are multifaceted and demand an extensive knowledge base. From treating traumatic injuries, managing critical illnesses, performing complex surgical procedures to providing emergency diagnostic services - these hospitals are equipped to handle a spectrum of veterinary emergencies. The expertise required to perform these tasks is driven by the principles of veterinary medicine, but is heavily influenced by technological advancements in medical equipment and techniques.

However, the operation of Emergency Veterinary Hospitals is not devoid of challenges. One such obstacle is the high cost of emergency veterinary care, stemming from the high operational costs these hospitals incur. These include, but are not limited to, costs for specialized equipment, medications, and highly skilled personnel. Further, the unpredictable and demanding nature of emergency cases may lead to mental and emotional stress amongst staff.

Despite these hurdles, the role played by these facilities in maintaining animal health is immutable. They provide timely intervention, thus reducing morbidity and mortality rates among pets. Moreover, they cater to the needs of animals outside regular clinic hours, thereby serving as a backbone of the veterinary healthcare system.

Additionally, there are economic and social implications. From an economic perspective, these hospitals contribute to job creation, not only within the hospital but also indirectly in sectors such as veterinary pharmaceuticals and equipment manufacturing. From a societal viewpoint, they contribute to animal welfare and public health, both of which have positive ripple effects on community well-being.

To borrow an econometric term, the operation of these Emergency Veterinary Hospitals could be seen as a "Pareto Improvement" in the veterinary healthcare system. In other words, they make at least one party (the pets) better off without making any other party (the pet owners) worse off.

In conclusion, the operation of Emergency Veterinary Hospitals is a fascinating interplay of veterinary science, technology, economics and societal needs. Despite the challenges, they continue to serve as an invaluable component of the animal healthcare system. The next time one finds oneself at the doors of such a facility, do take a moment to appreciate the labyrinth of complexity, knowledge and dedication that forms its very fabric.