Is there a location within Windows 8 where I can save a file and then be able to open it later on another computer in my network when the original computer is turned off?
1) Use SkyDrive. This is the easiest option. You could save your files to the SkyDrive to have them available on any computer that you sign in to the SkyDrive with. It doesn't require the source computer to be turned on since the files are store on SkyDrive. You could use the SkyDrive app (Metro - Windows 8) or SkyDrive desktop app (all Windows).
2) Use Offline Files. As long the network folder you create to share on both computers stay in sync, the files in it will be available on the other computer even when turned off.