Practically NO.

There are folks using dropbox and NAS devices, but those remote file systems always appear to lead to a problem that sees them posting in this forum.

Seriously though. Do you really need a gigabyte of mail at your fingertips on two devices?

" Do you really need a gigabyte of mail at your fingertips on two devices?"

That's a fair question. Is there a way to store all of that "important" old email so that it can be retrieved if needed?

You can move messages off of the server to a local archive on one computer. Either manually to 'Local Folders', or use the Archive function. https://support.mozilla.org/kb/archived-messages

Make sure to create a full backup of your Thunderbird profile folder prior to messing with it. https://support.mozilla.org/kb/profiles-where-thunderbird-stores-user-data#w_backing-up-a-profile

I came up with the 5gB figure by going in to my profile folder, finding the email folder, and getting the size from the "properties". After your replies, I dug a little further, and found that my empty drafts folder has, according to Windows, 248,094KB in it. Right now, I have 9 messages in my inbox, Which Thunderbird says total 409.7 KB. Windows shows inbox at 1,042,727KB.

How do I get an accurate measurement of my emails so I can make an informed decision as to what to do to solve my problem?

You should compact folders. http://kb.mozillazine.org/Compacting_folders

And don't forget to backup the profile before.