Windows Server 2003 is no longer supported by Microsoft. When I talk to customers, I still see organizations (even banks, retailers, and pharmaceuticals) using Server 2003 on their production networks. There are enough reasons to upgrade from Server 2003 and I am not going to explain them here. The same holds true even for AD DS. Sometimes, it is not easy to upgrade from one version to another, especially under a limited budget. But it's always important to evaluate the risks an enterprise will have without an upgrade. We need to be mindful of protecting the right stuff and investing in the correct areas of operation.

Windows Server 2003's forest and domain functional levels have been deprecated in AD DS 2016. The same happened in Windows Server 2012 R2: if you create a new domain, you cannot use a Server 2003 domain or forest functional levels. As shown in the following diagrams, you cannot deploy an AD DS 2016 domain with a Server 2003 domain and forest functional levels:

If your domain was originally created using Windows Server 2003 and you need to migrate to Server 2016, you can still do it. You can add a domain controller with Server 2016 and migrate FSMO roles over. In some documentation, I have seen people stating it's not possible to migrate from Server 2003 to 2016 directly, but that's not true.