For the past year I've been leaving My eBay up all day and refreshing occasionally. Sometimes I have to sign in again, sometimes not, but I always have to sign in more then two or three times a day.
And you're right -- "Keep me signed in" doesn't fix the problem.
I've also noticed that I have to sign in again more frequently when there's something eBay is trying to sell me on -- right now it's the new Feedback system -- which means an extra step in getting back to the page I was trying to refresh.