So, hypothetically, I could sit this UNIX box by my router and all configuration would be done via local SSH. Then once it's set up, I could SSH in from the internet and use UNIX (and all the tools contained within). Does something like this exist? If not, how could I build it? Is it possible to have just the UNIX OS (no GUI for example) installed on a computer?
The scenario you are describing is called a 'headless' install. Most server distributions of linux are setup this way.
For example, Ubuntu is released in 3 different flavors
The desktop edition comes with a Desktop Manager (Gnome, KDE, XFCE) which run on top of the X11 window system.
The server edition by default does not install a Desktop Manager, you just get the console.
Almost every enterprise server is configured to be headless. The only interaction with the server is performed over IPMI, a KVM switch, or SSH. Imagine how cumbersome it would be to plug a monitor into each blade in a datacenter one by one.
If you do not plan on using the GUI on your server, it is wise to turn it off. Leaving the GUI installed can consume unnecessary resources.
Additionally, it increases the attack surface of the server. Disabling the GUI is one of the items in Red Hat's official server hardening checklist.
Disabling the GUI
The easiest way to disable the gui is to change your run level. You can change it on the fly with the following command
To undo this use the following command
You can see your current run level by entering the command
To make the changes persist across reboots, edit the /etc/inittab file.
Change '5' to what ever runlevel you want (3 would probably be a good choice)
sudo vim /etc/inittab id:5:initdefault:
Sure, this is done all the time. I will assume that by "UNIX", you are accepting "Linux". I personally use the Slackware distribution of Linux. You can still install the GUI (X11) if you want and use it remotely through SSH tunnel, but that's not necessary.
My home server manages a RAID array, firewall, and a host of other services. It's just an old Pentium 4 box - nothing special, except that it has lots of hard drives in it =) I still have a spare monitor and keyboard lying around in case I have issues. You will need a monitor and a keyboard when you do the installation, but after you get networking going, you can run the machine without them.