Thanks for the tag, Rich...
survivalfoodsinc.com,
I apologize in advance for the length of my response. I wanted to make sure I left no stone unturned.
I definitely want to help get this sorted out for you. Of course, I don't know the specifics of your account, so I can only speak generally and advise based on what I expect is happening after having read your posts.
To start, you need to understand the basic structure of WordPress (and other dynamic web apps). When dealing with a migration/backup, you need two basic components - the hosted files and the database. Both of these are important; the site cannot exist without one or the other. The hosted files contain the basic files for making WordPress work, but they also include things like your theme, plugins, and uploaded images. The uploaded images are usually the most important hosted files as they are unique to your blog. Other than that, the site is fairly easy to rebuild, if absolutely necessary, because most people use publicly available themes and plugins that can just be reinstalled if necessary.
The database, which is stored separately from the hosted files, are the real meat of your site. Any posts you wrote, comments people added, settings you assigned, users and passwords, etc, are stored in the database. Pretty much anything that's unique to your site (other than uploaded images/files) are in the database, and that's why the others in this thread have placed such importance on it.
You mentioned that you have a copy of the files, so I won't discuss that in much detail. Simply put, though, you can backup the hosted files by connecting to your hosting account with FTP and downloading everything in there.
Backing up the database is different, and there are several ways to get it. At GoDaddy.com, we try to make it simple by having a "backup database" function in the Hosting Control Center. It results in a backup of the database as a single SQL file stored in a special folder in the hosting account (which can then be downloaded through FTP or our File Manager tool). We have a help article that details the database backup process - http://x.co/M1kf
WordPress also tries to make the migration process easy with an "export" option in the admin area. For many users, this is extremely useful, but it should be noted that only certain data (posts, comments, etc) can be exported. For a true backup, you really want to get a complete database backup as mentioned above.
Now, the real question for you is whether or not you still have access to the database. In other words, is your hosting at GoDaddy.com still alive? To be clear, it may very well still be active in the GoDaddy.com account even if you've taken steps to point the domain name elsewhere. Don't assume it's deleted just because you can't get to the site right now... If the hosting account is still active at GoDaddy.com, then you can backup the database. If the hosting account was cancelled, you won't be able to access the data because it's deleted immediately upon cancellation. Luckily, we can normally restore data from shared hosting accounts within 30 days of cancellation if you didn't make a backup of your own or if something happened to the one you made. Note, though, that we do charge a fee for this recovery.
You said that your login no longer works at GoDaddy.com, but it's not completely clear what you mean by that. Maybe the hosting account is gone, or maybe the person who originally set it up changed the credentials, or maybe there's just some confusion about what credentials to use (GoDaddy.com customer account versus hosting account). Once you have positively determined whether or not the original hosting account is still active, you'll know how to proceed. If it's active, you'll have to work on getting the proper credentials and going through the database backup process. If it isn't, you may be able to get it restored. If it's been too long, or you're not the account holder and can't get the proper credentials for the main GoDaddy.com customer account, you will, in all likelihood, not be able to recover the data. I hope that's not the case.
So, my advice at this point is to contact GoDaddy.com Support directly (again). Now that you're armed with the proper understanding of how a WordPress site works and the necessary conditions for recovering the data you need, I expect you'll have an easier time confirming what can and cannot be done.
Very important note - For security purposes, our representatives can't discuss specifics of an account over the phone unless you can validate certain information. This is a concern for me because you said someone else setup the account. If you can't properly validate, you'll probably want to ask for help from the person who originally set up the account. Assuming you can validate on the phone, our Support staff will be able to assist you with getting access online. Contact information for our Support teams is at http://x.co/M1p1 or you can just call 480.505.8877.
I sincerely hope you are able to get proper access to the account in question so your site can live on. Good luck to you.
^Alon