DNS Error 80710102 is an Internet issue with the Sony PlayStation 3. According to the Official Sony Support site, the error occurs when there is a timeout when the PlayStation attempts to connect to the Internet Service Provider's Domain Name Server.
One of the more likely causes of this DNS Error is that the PlayStation cannot properly discern the DNS. In this case, if the DNS appears as 0.0.0.0 then it can be manually entered. The DNS can be acquired from the service provider, located somewhere on the wireless router, or found using command prompts. According to Sony, frequent causes of this error are a nonworking Internet connection (which requires a fix unrelated to the PlayStation), having PlayStation network settings unintentionally reconfigured, or DNS-related issues. There are also a plethora of public DNS that can be used to connect to the Internet and avoid DNS errors. For example, the public Google DNS is 126.96.36.199 preferred and 188.8.131.52 alternate. Public DNS has the advantage of only being dependent on a stable Internet connection and not relying on the ISP and proper network configuration. Finally, third-party sites and tools can be used to determine the DNS address of specific ISPs and sites.