Is it possible to make a paper wallet out of a bitcoin-qt wallet? I have a single bitcoin address with all my savings on it. I want to make a paper backup for precaution (as if my hard drive somehow exploded) with that specific address. Is that possible?

Keep in mind that I want a physical backup but at the same time I want to be able to manage the wallet from bitcoin-qt.
Right now, that's not supported. You can try (see user2194702's answer), but if you don't know the exact workings of the wallet, you're very likely to shoot yourself in the foot.

For example, Bitcoin-Qt will send change to a new address every time you create a transaction, and in general does not follow the "balance per address" concept but a "balance per wallet". This improves the privacy of the system (for you, but also for others), but complicates backups. In particular, you cannot backup a wallet by just making a backup of a single key.

HD Wallets (hierarchical deterministic wallets, see BIP32) are intended to overcome this problem, by making sure you can backup a wallet by only making a backup of a single seed that is used to derive all keys. This is planned to be implemented in Bitcoin-Qt, but not yet available.