Is your browser set to erase cookies when closed? That will cause you to have to sign in every time you start a new session.
Similar problems can be caused by using privacy software such as Ghostery or Disconnect, or by blocking tracking, blocking scripts or blocking third party cookies.
If you have a corrupt eBay cookie, manually erasing your eBay cookies once might help.
Trying another browser might allow you to narrow down the problem to your settings in your current browser.
