Repeated Captcha requests or failures might be a sign that your browser is having problems saving cookies properly, or else a sign of a network issue. Using a VPN or a hotspot, or certain kinds of privacy software might also cause some issues.
Make sure your browser is up-to-date and has JavaScript enabled (and not blocked) and that cookies are not being blocked:
https://www.whatismybrowser.com/
Try using a different browser or a private or incognito browser window and see if you run into the same Captcha issues. If not, try manually erasing just your eBay cookies manually in your original browser.
Try an alternative link to sign in to rule out an issue with a particular link:
https://signin.ebay.com/ws/eBayISAPI.dll
If you are using a mobile browser, try changing to a different browser mode: mobile, classic or desktop. Try a desktop browser or a mobile browser in desktop mode.
Or try using the eBay app.
