Safari quits for no reason after a period of non use. I am running v 10.9 mavericks. It did this with the last version of the OS, too.

Try troubleshooting Safari extensions and third party plugins.

From the Safari menu bar click Safari > Preferences then select the Extensions tab. Turn that OFF, quit and relaunch Safari to test. If that helped, turn one extension on then quit and relaunch Safari to test until you find the incompatible extension then click uninstall.

If it's not an extensions issue, try troubleshooting third party plug-ins.

Back to Safari > Preferences. This time select the Security tab. Deselect: Allow all other plug-ins. Quit and relaunch Safari to test.

If that made a difference, instructions for troubleshooting plugins here.

Do you have anti virus software installed ??

Or an app called MacKeeper ??

That sounds like Automatic Termination

Also see Adopt Automatic Termination (...a facility introduced in OS X Lion. It allows underlying processes to be terminated by the system automatically, independent of the running state of the application itself. We intend to rely on Automatic Termination even more in the future, and encourage you to adopt it in your applications....)

I don't know that this is automatic termination, unless you typically leave Safari running with no open windows. All apps I'm aware of that use automatic termination, such as Preview and TextEdit, only quit automatically when they don't have any open windows.

If you always have a window open in Safari, it shouldn't quit. I typically keep Safari running non-stop for days or even weeks without quitting, and I've never had an issue with it quitting on its own. So, in such a case, I think there must be something else going on. Perhaps a low memory situation could cause Safari to auto terminate even with open windows, or perhaps there's some internet plug-in installed that is causing the problem. You've already checked for extensions, but check for plug-ins as well. Look in:

~/Library/Internet Plug-Ins/

(If you're not sure how to find that, see Locating files from paths.)

Anything there can be at least temporarily removed for testing. Also check:

/Library/Internet Plug-Ins/

That one will normally contain the following items:

Default Browser.plugin

nsIQTScriptablePlugin.xpt

Quartz Composer.webplugin

QuickTime Plugin.plugin

Anything else can be removed for testing.