Is it possible to bring all current and historical emails from an eircom.net account to my new gmail

You can move what's in your inbox quite easily - see this Irish Times article:

Doesn't sort out your sent emails though. There is a very clear article from 2016 here on how to connect up a Zimbra (eircom webmail) and Gmail account to move other folders but sadly it no longer works. I suspect it may work with a Gsuite account but not a gmail.com one. I tried every possible option of 2-factor authentication, app-specific passwords, turning on less secure app access, different imap ports and got *nowhere* trying to do this for my friend's gmail.com. If I can get it working with my GSuite account and manage to transfer her sent emails that way, I'll post again here.

Beware there are a lot of paid apps online that claim to convert Zimbra exported TGZ files to gmail. I don't see any evidence that any of them actually work, but I could be wrong. Good luck, if you find an answer for the sent emails, please post it!

Imogen

You then need to add to Thunderbird the IMAP accounts for both your old eircom.net account and your new Gmail account. Here's a help article about adding email accounts to Thunderbird.

This will just set up POP email, and we then need to change that using Tools>account settings. The required incoming settings for this process for your old emails (which is temporary, just for transferring your old emails, so you will be able to delete the whole Thunderbird account afterwards) are:

IMAP, with webmail.eir.ie as the incoming server, incoming port 143 but no SSL

and that means then you will then need to confirm to Thunderbird that you understand that it's not very secure. (but you're deleting it shortly so should not be too much of a concern).

Don't worry about the outgoing sending server which won't work if you are not using an eircom internet connection. Just let Thunderbird choose what it likes for sending from eircom.net - we aren't sending or receiving, we're using this setup to move emails from one account to the other.

***important extra step*** Thunderbird needs to know it needs to "subscribe" to any other folders of email you want to transfer to Gmail, such as your sent folder which is described in the attached screenshot to this post, taken from https://wiki.zimbra.com/wiki/Accessing_Zimbra_Collaboration_Server_with_Thunderbird - most of that article is not relevant, but the bit about subscribing is.

Add a second account to Thunderbird with the details of your Gmail (start at Tools>Account settings). It will want your Gmail address and password.

It will probably take rather a long time to download all the emails!

Now, because both accounts are using iMap, you will be able to copy emails from one account to the other, for example, from the Eircom sent folder to the Gmail sent folder. To select all the emails in an eircom.net folder, click it first then use ctrl-a or cmd-a if on a Mac. Then right-click and choose "copy to" whichever folder in the gmail account you wish to transfer them to. 

Be patient if you have a lot of emails! When you've finished, be sure to log into Gmail by whatever your normal route is, to check everything is where it should be. Then don't forget to either remove the eircom account from Thunderbird, or delete the whole Thunderbird app.

Now you can tell Eir where to stuff their new €5.99 / month fee. Please let other people know this free route, as I see lots of dubious apps charging $99 to supposedly do a Zimbra/eircom to Gmail conversion. Good luck!