Why, when I go to my beay account, do I have to login when I repeatedly tick the stay logged in box. Every other website, such as this understands that so I guess my security settings etc are correct.
Actually thats not true. My orange webmaail account has stopped keeping my settings a while back but there is no time relationship between the two.

something deleting the cookie?

eBay only keeps you logged in for 24 hours.

Ebay prompts you to re-login for some actions IIRC? It’s a while since I used it though so I’m not 100% sure. Amazon does a similar thing, I can logged in with cached credentials but if I want to buy anything it prompts me for my password again.

Does it do the same thing in a different browser?

You can use this Greasemonkey script with Firefox to login to ebay automatically. http://userscripts.org/scripts/show/1766