I know, this problem can be faced by so many people and many of them have uninstalled and re-installed the sql server for resolving this issue. In my observation the problem of not connecting the database service locally is just because of your network connection you are using, in most of the cases these problems will come when you are using wi-fi network.
Solution is, if you are using wi-fi then just right click on status of the network and get the ip details and enter the same ip in sql server name, it will work. Regards Vishwajeet