With the command line client, enter
bitcoind dumpprivkey youraddress
This will output the private key for "youraddress". I don't know how you'd do that with the qt client.
The keys are stored in your wallet.dat. If you back up wallet.dat there's no need to keep separate copies of the private keys. Unless you want to print them on paper and store them in a bank safe or something.
Encrypt your wallet.dat and unlock it only when you want to spend coins.