No servers? That's right. It's now possible, and with the same (arguably much better) services.
You can maintain all the necessary services including things like:
Identity Management (Like, when you log in to Windows)
Accounting, HR, and Other Functions
Software and Patch Management
Anti-virus and Email filtering
All of this can be offered with no servers in your building. It's a game changer in how to manage information systems and will be hugely beneficial once the world fully embraces it.
The last big hold-out would have been identity management. This is used when it is necessary to centrally control access to various network resources. It's definitely still a necessity for most organizations, as the backbone of the system. The solution most organizations use is Microsoft Active Directory. They have recently made available a cloud-based version of it inside their Azure service.
Azure is an unfortunately mysterious sounding service. It is enormously capable, and so, at first glance, it appears to be only for giant organizations. This is not the case, at all. It is the replacement for you IT closet. Everything you run in your IT closet, can live inside Microsoft's data centres. Everything.
Once this becomes clear, your whole approach to IT spending changes. Suddenly you will realize that servers, power supplies, racks, air conditioners, space, etc, have no place in your organization! Get rid of them! Save yourself the risk, money and headaches. The incremental increase in focus it will bring to your organization will be well worth it.