Solar Flare wipes all hard drives... What happens? Imagine, for a moment, that a massive solar flare wipes out all hard drives on planet Earth.  All hard drives are wiped clean and with them all records of the Bitcoin blockchain.

Would we be able to recover?  Is there some alternate backup of the blockchain?  Do we hope that at least one computer is protected beneath ground?

I assume everyone's personal wallets would be lost unless they used paper wallets.  Certainly a big blow to the Bitcoin community.
All hard drives are wiped clean and with them all records of the Bitcoin blockchain.
Would we be able to recover?

Recover what? You stated all hard drives are wiped clean, so obviously we would not have a backup to recover the network from.

Is there some alternate backup of the blockchain?

Perhaps someone has printed the blockchain out onto paper. In an event like the one you've described, we should hope that there are many copies of this paper format blockchain, so that we don't have to trust that one person to have provided us with a copy of the blockchain history.

Do we hope that at least one computer is protected beneath ground?

If there was, then your premise of all hard drives being destroyed would be violated. Same as above, we should hope for many independent copies to surface, stored underground (or whatever) all around the world.

I assume everyone's personal wallets would be lost unless they used paper wallets.

Even if you used a paper wallet, if every record of the blockchain history was destroyed, your paper wallet would be useless. A paper wallet simply holds the private key capable of spending the bitcoin stored at it's corresponding address. If the blockchain record were lost completely, there would be no network to agree upon how much bitcoin is stored at that address. So a paper wallet that held 100 BTC before the catastrophe would be no different than one that held 0 BTC. With no blockchain to make a claim against, they would both be effectively worth 0 BTC.

Certainly a big blow to the Bitcoin community.

Sure, but any event catastrophic enough to knock the bitcoin blockchain out would be devastating FAR beyond that. Bitcoin is highly distributed, to destroy it would take an event of unbelievable magnitude. While I believe Bitcoin is important, there are many other systems more critical to our everyday life that would be affected also.