Quitting and reopening Mail loses all accounts
Chase reports and asks:

Also, the folder ~/Library/Mail only has two folders
Mail Lost+Found
V2
All of my mailboxes are contained within V2... is that normal?

Those folders should exist, but at least based on the contents of my own ~/Library/Mail folder, it's incomplete. I also have individual folders for each of my email accounts and those folders contain mbox files which I assume are the local copies of the IMAP mailstore.

By way of troubleshooting, it might be interesting to know what the file permissions are for the Mail directory.
Open the terminal, and type (or cut and paste) the following command:
ls -l ~/Library/Mail

The results (edited here to show only the relevant line) for my own functional Mail folder are:
drwx------    13 jaberg  staff    442 Oct 13 00:25 Mail

The permissions basically tell us that this a directory and that the owner, jaberg (me) is able to read, write and execute. (The execute permission of a directory is sometimes thought of as the search permission. You need the x in order to be allowed access to the information about the contents of the directory.)
You can also view permissions from Finder by selecting the folder and performing Get Info… from the file menu (or ⌘+i), but you won't get as much information.

You'll find more information about accessing and interpreting file permissions in the Apple Knowledgebase: Troubleshooting permissions issues in Mac OS X
If you're permissions differ from those shown here, you can try to Repair Permissions using Disk Utility (this doesn't require a restart or booting from a different volume) or, if you're comfortable, change the permissions of the Mail folder to match those shown using the chmod command. You can learn about chmod from its man(ual) pages, accessed from the terminal by typing:
man chmod