Trying to access my BTC from Bitcoin-Qt wallet that is out of sync I bought BTC and sent them straight to my Bitcoin-QT wallet. Now opened my wallet (ver. 4.8.3) and it says ~29000 blocks remaining to synchronize but this is not happening even though I have 14 active connections.

I have tried accessing my Private Key through both Help>DebugWindow>Console and the version using CMD bitcoind dumpprivkey etc with both returning the error message "Safe mode: Warning: Displayed transactions may not be correct! You may need to upgrade, or other nodes may need to upgrade. (code -2)"

I don't have an encrypted wallet and just want to have access to my money.
Assuming you mean 0.8.4 ? (Or 3). Latest is 0.8.5 now but nevertheless...

What is the fastest possible way to download the blockchain?

Although I wouldn't recommend the bootstrap.dat answer because you'd need to create a new wallet for that--and eventually would likely have to resume syncing the old one anyway.  

Edit: it appears I'm mistaken about the above paragraph.  The thing to know about bootstrap.dat is your client still has to verify everything that was imported over the network so it's still going to take some time. But a combination of both answers might be best. And only download from a trusted source. 

Try to just get the fastest connections as per the "high bandwidth" answer.  There's really nothing much you can do in the meantime because wherever else you try to send the coins is not going to go through until the entire block chain is synced. 

I've gone through this when I had to re-download the whole thing after a kernel panic -- LFMF -- don't import the private keys elsewhere before it's synced.