If you already have a git repository in place, first copy all your files to that directory (yes, manually) and do a git commit -a to commit all the files into the repository. I'm assuming here that you've already initialized the repository at C:\Users\username\GitHub\project_name\ with git init.
In eclipse, go to File → Switch Workspace → Other... and point it to a workspace of your choice that can be completely different than the location of your code, or your earlier workspace. In fact, don't point it to the directory which contains your git repository.
Once you have a clean workspace, go to File → New → Other..., select Git → Git Repository and enter the path of your git repository (C:\Users\username\GitHub\project_name\). Enter a name for the repository, and click Finish.
I would really recommend you read at least the first few chapters of the git book to understand how git works, and to help you push and pull code to and from remote repositories.
If your existing repository is not git, you're going to have a hard time keeping the directories in sync. You might want to setup rsync to sync the directories. There is no way AFAIK for eclipse to automagically keep the two repositories in sync.
Answer from Anand on Stack OverflowVideos
You need to tell EGit where the default system configuration is for it to be able to pick up the values. There's a Browse button which should allow you to change it according to the help pages:
https://wiki.eclipse.org/EGit/User_Guide#Pointing_out_the_System_wide_configuration
You can also set this property in the User Settings which Eclipse will respect. Global is 'for all users of this machine' and User Settings is typically stored in your home directory, i.e. 'for just this user'.
You'd have to find where the file that's being created by git config --global on your setup is. It's likely that git isn't on the path, and that's why Eclipse can't auto-guess where it should be.
By the way, unless you really, really need to have CRLF on the files you should prefer to keep them as how they are stored remotely - it will be faster for EGit and Git to perform deltas if it doesn't have to do line-ending conversion on most operations.
I had a similar problem so let me add this for clarification: in Eclipse Preferences > Team > Git > Configuration the System Settings Tab was empty and "Location:" said "Unknown". The EGit User Guide (https://wiki.eclipse.org/EGit/User_Guide#Pointing_out_the_System_wide_configuration) tells that "If you selected one of the options to use Git from the Command Line Prompt when you installed Git for Windows, then the location of the system wide settings is filled in with a path and everything is fine. If not, use the Browse button to locate where Git is installed, e.g. C:\Program Files(x86)\Git." Indeed I did not select the option to use "Git from the Command line ..." when I installed Git. But instead of a "Browse" button my Eclipse 2020-09 has an "Open" button and that one was GREYED OUT, so I was unable to tell EGit in the above mentioned "System Settings" Tab where the local Git installation could be found. Furthermore I did not know which path precisely had to be added to the systems PATH variable. I then reinstalled Git with the option to use "Git from the Command line ..." and found out that "C:\Program Files\Git\cmd" was added to the System PATH. After a restart of Eclipse the "System Settings" Tab had the "Location" configured as: "C:\Program Files\Git\etc\gitconfig" and all the missing keys and corresponding values were set.