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I am closing out an email account. But I need to be able to save each email for potential future reference. What is the simplest way I can achieve this?

First figure out what kind of an email account it is: POP3, IMAP, or MSX (It's probably one of these three).
If it's MSX, 1) I'm sorry, and 2) I'm not sure how to help you.
If it's one of the other two, there are many email clients that support downloading the hierarchy of your email account onto your computer through their settings, but you will need to make sure you have enough space to hold all of everything before you begin, otherwise you're gonna have problems.

--Patrick

Oh I was hoping a 4gb stick would be enough.

...and it might be (text is pretty compact), unless you have a bunch of attachments and pictures and stuff that'll unexpectedly balloon the size.

--Patirck

That worried me. I don't need the images in the signature of the emails but do want to keep the attachments.

Since image signatures are often technically "attachments," I don't know how that would work.
Did you find out whether you are POP/IMAP/MSX yet?
...do you know where to look?

--Patrick

I am mobile presently. This will be a project for the near future. Just learning more before I begin.

Then you will need to determine how you would access that same account on a computer. This is something you would check with your provider to learn their "SMTP server and port settings." You don't need to know what SMTP stands for, but that is the key question you would pose to the Internet, to the support staff, or whomever you ask for the info. Anyone who knows the answer to that question should understand what you want.
Once you know the answer to the above question, you have 90% of everything that you will need to get and save all your email onto a computer, after which you can move it to whatever format you wish.

--Patrick

On actual PC now,

One account is POP and two are IMAP. (trouble is I have one entire account to save and some emails from another account to save as well)

Using Thunderbird as a client.

How to archive emails using Thunderbird.

--Patrick

Thanks got them all saved now down to a few gb.