I am using the default config while adding the specific directory with nginx installed on my ubuntu 12.04 machine.

server {
        #listen   80; ## listen for ipv4; this line is default and implied
        #listen   [::]:80 default ipv6only=on; ## listen for ipv6

        index index.html index.htm;

        # Make site accessible from http://localhost/
        server_name localhost;

        location / {
                # First attempt to serve request as file, then
                # as directory, then fall back to index.html
                root /username/test/static;
                try_files $uri $uri/ /index.html;
                # Uncomment to enable naxsi on this location
                # include /etc/nginx/naxsi.rules
        }
...

...
}

I just want a simple static nginx server to serve files out of that directory. However, checking the error.log I see

2014/09/10 16:55:16 [crit] 10808#0: *2 stat() "/username/test/static/index.html" failed (13: Permission denied), client:, server: localhost, request: "GET /favicon.ico HTTP/1.1", host: "domain"
2014/09/10 16:55:16 [error] 10808#0: *2 rewrite or internal redirection cycle while internally redirecting to "/index.html

I've already done chown -R www-data:www-data on /username/test/static, I've set them to chmod 755. I don't know what else needs to be set.

Nginx operates within the directory, so if you can't cd to that directory from the nginx user then it will fail (as does the stat command in your log). Make sure the www-user can cd all the way to the /username/test/static. You can confirm that the stat will fail or succeed by running

sudo -u www-data stat /username/test/static

In your case probably the /username directory is the issue here. Usually www-data does not have permissions to cd to other users home directories.

The best solution in that case would be to add www-data to username group:

gpasswd -a www-data username

and make sure that username group can enter all directories along the path:

chmod g+x /username && chmod g+x /username/test && chmod g+x /username/test/static

For your changes to work, restart nginx

nginx -s reload