The Internet uses Domain Name Service (DNS) to map domain names to IP addresses. Therefore, even if you don't use DNS internally, you must provide DNS name servers on the Internet in order to connect your network to the Internet. You should have at least two name servers, a primary and a secondary server. For robustness, the secondary server should not be connected to the Internet through the same gateway as the primary server. Since many organizations are not big enough to have multiple gateways to the Internet, a common solution is to make a reciprocal arrangement with another organization to provide secondary name service for each other.
If you are connecting to the Internet through an Internet service provider, they may be able to provide name service for your organization, or help you locate someone to provide secondary name service if you are able to provide a primary name server.