Some general sign-in/verification suggestions:
Try another browser to rule out browser issues, preferably the latest version of a browser you have not tried already, such as Firefox Developer Edition or Chrome Beta, with default settings and no add-ons installed.
Try another sign-in link to rule out problems with an old or outdated link, particularly if you are using a bookmark.
https://www.ebay.com/signin/?sgn=reg&siteid=0
https://signin.ebay.com/ws/eBayISAPI.dll
There are many things that can interfere with Captcha verification.
Sometimes when eBay changes the sign-in process, it causes problems for some users; sometimes those problems are due to using an old browser or obsolete cookies (though not always). Some users are just annoyed by verification; while others actually get stuck in a loop that just keeps presenting new sign-in pages or new verification pages, or just will not let them sign in at all.
Make sure you are allowing the sign-in page to load completely before attempting to sign in -- users have indicated that attempting to sign in before the page has loaded can cause problems.
Make sure your browser is set to accept cookies. Try another browser or another device, or a private or incognito browser session. If you delete your cookies, you will likely have to sign in and go through strict verification again, but if you have problems with a blocked or corrupt cookie, manually deleting your eBay cookies can help. Some browsers seem to require more verification than others; try signing in and out more than once to see if it requires verification the second time around.
Make sure your browser has javascript enabled:
https://www.whatismybrowser.com/detect/is-javascript-enabled
VPNs can also cause sign-in issues, so try disabling yours if you use one and are encountering problems. Privacy add-ons like Ghostery or Disconnect can also cause issues.
Blocking tracking cookies or third-party tracking can also interfere with verification; see if you can disable that feature on the sign-in page or on the verification page.
If you are able to sign in, to stay signed in there is a "keep me signed in" checkbox on the sign-in page, but there is another checkbox in your account site preferences -- it might be worth checking that setting, or unchecking, saving, rechecking and saving again. That is, if you want to stay signed in and avoid having to go through verification.
If you have multiple accounts and must sign in and out to switch, consider using a dedicated browser for each account to minimize signing in and out. A private or incognito browser session is handy if you just need to check another account without signing out of your current account.
If you have successfully signed in using Google or Facebook before, try that again -- the sign in process using those may have different dependencies and may work better for you. But if you have not signed in using those before make sure you do not accidentally create a new eBay account when you attempt to sign in.
If you have the eBay app on your phone, you might find it easier to sign in using the eBay app.
