Your screen capture indicates your problem is not a Home <> Chromecast connection problem but rather something relative to your home network and connectivity with online access, or more specifically according the wording in the error message itself, with Google. Apparently your Chromecast is just an end-result so focusing more on the source of the problem, possibly your router and/or your broadband connection:
-- What kind of router do you have, an ISP-supplied modem/router unit or do you have your own separate router? If your own, what model?
-- Who is your ISP?