I backed up my private keys this way: Opened my peerunity wallet (bitcoin-qt fork), opened Debug window, issued dumpprivkey and I got my private key. Then, I wrote it on a piece of paper (!). Now, I would like to recover it. The wallet was encrypted at the time of dumping priv key. I have the key I used to encrypt it. Is privkey on my paper encrypted somehow? Peerunity is not accepting it when I'm trying to importprivkey. (tried it repeatedly)
I'm not familiar with Peerunity but here is how things work in bitcoin-qt, which is probably similar unless Peerunity changed something.
importprivkey accepts exactly the same format output by
dumpprivkey. The dumped private key is not encrypted in any way (if the walled was encrypted, you had to give the passphrase before dumping so that it could be decrypted).
The most likely explanation is that you wrote down the key incorrectly. Private keys usually include a checksum so that errors can be detected immediately, and that seems to be what is failing here. It shouldn't matter that the wallet is out of sync; importing a key doesn't need access to the network (though you will need to sync with the network to get an up-to-date balance).
If you made a mistake when writing down the key, you may be able to do something similar to the approach in the question Can you help me fix this address? to try changing a few characters at a time in hopes of recovering the correct key.