Where are by Thunderbird gmails stored?

It depends. What is your account type - POP or IMAP?

I am running two email server programs right now (Outlook and Thunderbird).

Neither are server programs, they are email clients running on your local computer.

I do not want TBird to store all emails on my hard drive but I am told they are on my hard drive.

What exactly are you being told?

If so this will eat up ALL my disc space.

Google offers 15GB of free disk space which can be used for mail, and other data. If you don't have that much free disk space on your computer you'll probably have other problems. Get a new disk, or even a new computer. Disk space is not expensive.

Check the 'synchronization&storage' pane to manage this. Messages are stored on the server, and Thunderbird keeps on disk according to the specs. The header for each is there, but you can keep sync to just the most recent. You can also set which folders should be kept for offline access.

Thanks but cryptic reference to a needle in a haystack is not helpful but just sends me down a rabbit hole. Please communicate clearly and completely. Nowhere do I see any reference 'synchronization&storage'. What are you talking about?

P.S. Your answer appears to conflict with that my Tbird screen shows. That screen says Tbird is "downloading" the messages. If so where is it downloading them to. Are these not saved to my hard drive?

The access to messages is dependent on settings for each account. As such, synchronization&storage is part of an account's settings. If you look at the account setting pane, you will find it. Yes, messages are downloaded for local access, but storage is determined by those settings. Thunderbird allows for messages to be saved locally for offline access. You can adjust the sync level to manage that.

Thanks but cryptic reference to a needle in a haystack is not helpful but just sends me down a rabbit hole. Please communicate clearly and completely. Nowhere do I see any reference 'synchronization&storage'. What are you talking about?

If you go into Account settings, you will find what David is talking about. See https://support.mozilla.org/en-US/kb/configuration-options-accounts

Your reliance of a status line for your information is not helpful here as even if no mail is actually stored in Thunderbird, some header information still has to be downloaded so it can show you things like your message lists and will be shown a downloading message on of whatever number of new mail is present to receive.

I suggest you invest some time reading these support articles. https://support.mozilla.org/en-US/kb/difference-between-imap-and-pop3 https://support.mozilla.org/en-US/kb/imap-synchronization