Using Bitcoin, you can have transactions sent to different addresses. These addresses are, in turn, contained in wallets accessed by keys. The downside is, if you lose the keys, you will lose the Bitcoins (purchasing power) at those addresses in those wallets.
What constitutes a widely accepted and reasonably safe backup system for the keys? Are there means of storing or "hosting" the keys in a way that most authorities would agree has minimal risk of loss?
The most common approach seems to be backing up your (encrypted) wallet on cloud hosting services such as Dropbox and Google Drive. To reduce the risk of theft, use a very strong password to encrypt your wallet, and do not store the password anywhere less secure than the password itself (e.g. if you use a password manager, it should also have a very strong password).
To reduce the risk of accidental loss, keep backups in multiple places. E.g. Dropbox, Google Drive, email it to your family and friends, back it up to an external hard drive; if one of these disappears, chances are still good that you can recover it from another source. You don't want the keys to be in only one physical location...e.g. having the only backup be in your house means that a house fire or theft could mean you lose your bitcoins.
Just don't forget the password! Maybe you should write it down and keep it in a safe deposit box at a bank, and/or keep paper wallets there, if it's worth the cost to you (maybe $15/year for a small box).
A rather different approach is taken by deterministic wallets like Armory and Electrum, which generate one root key or seed and then generate your private keys and addresses from that. Armory's root key works on the idea that you print out the ~256-bit root key in one or more fragments, and need the appropriate paper records to restore your wallet. Electrum's seed is a 128-bit number represented by 12 common words, which you commit to memory and/or write down. Securing these records would follow similar guidelines to a written password or paper wallets.
Just like cash, if you want complete authority over your money, you also have to be responsible in storing in a way so you don't lose it or get it stolen.
The Bitcoin protocol has ways to rely on third-parties to distribute the responsibility, that regular cash doesn't. The idea is that of multi-signature transactions. The bitcoins can only be claimed by 2 out of 3 keys. Instead of a single key, as most transactions are currently done, 2-of-3 transactions are much more secure from loss or theft.
In the normal scenario, if you lose the key, you lose all your bitcoins. In this scenario, you have to lose 2 out of the 3 keys to lose your bitcoins. Ideally you would store 2 keys and the third would be in the hands of an untrusted third-party. If you lose one of the 2 keys, you can still rely on the third-party to claim your funds.
In my opinion this provides an incredible extra security measure that no other currency medium has ever been able to provide before: the ability to distribute responsibility of safekeeping your money, without having to trust anybody.
