I can't send emails on my gmail a/c through iMac it shows gmail offline

Please read and methodically follow Apple's Mail Troubleshooting article.

The Apple instructions didn't work. Anyone actually experienced it who could help?

You may have a corrupt preference even though it looks fine. Enter Mail - Preferences - Accounts and re-input each preference even though it may look okay. If that does not work, create another user account (call it TEST Account) and temporarily create the same mail account and test it on that account. If the issue does not occur in the TEST account then you know the problem is specific to your user account.

Thanks. The trouble is that I'm so untechy I have no idea how to do that. Is there a step by step instruction somewhere?

How to do what, check the preferences? If it's that I already provided instructions, if you have difficulty with that then you may benefit by calling AppleCare or visiting a Genius for an appointment.

If you are referring to creating a new user account please look over OS X Mavericks: Set up users on your Mac,

It seems to me, as a non-techy person, that you have told me what to do but not how. That's what I mean. For example, I don't know how to "re-input each preference" or how to "create another user account (call it TEST Account) and temporarily create the same mail account". I didn't set it up originally so I am really starting from the very beginning. It is probably too much to ask of this forum.

I Recommend either a call to AppleCare or making an appt wirh a Genius.

Good luck.