I have seen these previous issues:
Keep getting logged out of StackOverflow today
Keep getting logged out of Chrome dev
Why do I keep getting logged out of SO and sister sites?
These were all back in 2010, and all are marked completed or decline. One issue was cookie-related that Jeff stated was fixed. Another was cookie-related but user error (they inadvertently blocked all 3rd-party cookies).
I know I haven't changed my cookie settings - and this symptom does not occur on dba.se or here on meta, just on the main SO site. It seems to be happening around hourly, but I'm not active enough today to confirm that it is every hour.
I'm using Chrome 23.0.1271.95 on Mac OS X 10.8.2, have tried disabling all of my plug-ins, have re-launched Chrome and rebooted, still same symptoms.