Bumping for interest, same thing started happening to me today. Been on Mavericks almost since day one with no problems, and I didn't install, delete or modify anything between yesterday and today that would cause this to happen. It appears that it's offline or the address is wrong, but I've been using Mail for years with the same settings, haven't touched it. Here's some relevant log entries, it simply appears to be the wrong address or offline.....
1/27/14 8:36:40.850 PM Mail: Metadata.framework [Error]: couldn't get the client port
1/27/14 8:28:51.240 PM Mail: [LogSocketErrors] Connecting failed for socket: 0x6000004cfab0, error: Error Domain=NSPOSIXErrorDomain Code=65 "The operation couldn’t be completed. No route to host"
1/27/14 8:28:38.117 PM icbaccountsd: -[ICBLocalDictionary discoverLocalIAAccounts]: Unknown high level account: com.apple.account.mail - F39B13FD-F33F-4EB9-A780-B6495A4C8E94
1/27/14 8:26:36.426 PM com.apple.preferences.internetaccounts.remoteservice: An instance 0x7fcc4c45cfb0 of class IMAPMailbox was deallocated while key value observers were still registered with it. Observation info was leaked, and may even become mistakenly attached to some other object. Set a breakpoint on NSKVODeallocateBreak to stop here in the debugger. Here's the current observation info:
<NSKeyValueObservationInfo 0x7fcc4c45d0d0> (
<NSKeyValueObservance 0x7fcc4c45d060: Observer: 0x7fcc4c45c8c0, Key path: uidNext, Options: <New: NO, Old: NO, Prior: NO> Context: 0x7fff89c8f44b, Property: 0x7fcc49f947f0>
1/27/14 8:13:50.225 PM sandboxd: () Mail(744) deny file-issue-extension /Applications/System Preferences.app
1/27/14 8:13:11.434 PM WindowServer: common_reenable_update: UI updates were finally reenabled by application "Mail" after 4.08 seconds (server forcibly re-enabled them after 1.00 seconds)
1/27/14 8:13:11.248 PM Mail: Metadata.framework [Error]: _MDAccessCopyClientPort failed retrying
1/27/14 8:13:07.846 PM Mail: Metadata.framework [Error]: MDPlistContainerRef _MDCopyCompleteFlatSchema(int): couldn't fetch complete schema from server ((ipc/send) invalid destination port)
1/27/14 8:13:07.287 PM Mail: No more available unread queries. Unread query not created for <MFMailbox:0x7fec2252bce0 (relative path=Gray, attributes=0, displayName=Gray uuid=flag-6)>. Setting unread count to 0.
1/27/14 8:13:07.286 PM Mail: Metadata.framework [Error]: Failed to get task count with error (null)
1/27/14 8:13:02.143 PM sandboxd: () Mail(744) deny file-write-create /Library/Mail/Bundles (Disabled)
I also disabled Little Snitch and the built in OSX firewall, tried pinging all the smtp and imap servers (they're up), rebooting, hitting the computer while swearing loudly....no dice.