Disabling ad blockers, pop-up blockers, script blockers or other privacy tools on the sign-in or verification page can help; many of those have settings that can interfere with verification. You may be able to turn them back on once you have signed in or checked out and updated your eBay cookies.
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 cookies manually in that case.
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.
Try a different payment method; try checking out directly on ebay.com without using PayPal.
Or try checking out using the eBay app or the browser on your phone.
