Disabling ad blockers, pop-up blockers, script blockers or other privacy tools on the sign-in or checkout page can help; many of those have settings that can interfere with checking out. You may be able to turn them back on once you have completed checkout and updated your eBay cookies.
In some cases, pages load better with an ad blocker or script blocker, though.
If you are using Firefox, try disabling Enhanced Tracking Protection on eBay using the shield icon to the left of the page address.
If you are able to sign in and check out using a private or incognito browser window, that might indicate a problem with a corrupt or blocked cookie. Try erasing just your eBay or PayPal cookies manually in that case.
If you are routinely clearing all your cookies, you will likely have to go through verification again each time you sign in.
Or try a different browser -- preferably the latest version of a popular browser that you do not normally use and one that has no add-ons or extensions installed.
Check your browser for updates here:
https://www.whatismybrowser.com/
Other possible things to look at:
If you are using PayPal's 2-factor authentication, temporarily disabling that may allow you to check out.
Another possible route if you are getting stuck signing in to PayPal -- open a separate browser tab, go to PayPal, use the gear icon to reach settings, select the Security tab, then use One Touch auto login to allow your browser to be recognized. You can disable the feature after checking out.
While you are in your PayPal account, be sure to check the summary page to see if PayPal needs any additional information or documentation from you; sometimes you may need to agree to new terms before you can proceed to checkout, and the default popup may not appear if you are checking out on eBay at the time.
