I agree with @Mikensan, I use Nas4Free on a computer very similar to yours. (I think I technically have 4GB of RAM.) Works like a charm. Nas4Free broke off from FreeNAS back in like 2010-2011 (nobody check those dates they could be terribly incorrect ) A simple install and a few quick clicks, and you should have a working NAS that is ideal for backups.
I am a little paranoid about backing up data, so I also use the rsync features in Nas4Free to create a redundant copy of my critical data to a mounted USB drive.
Before my Nas4Free setup, I was just using a standard samba file server configured on Ubuntu Server. Worked really well, but Nas4Free is definitely a step up for me.