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Backing up private keys in a secure manner is not straightforward. Once a backup has been compromised you need to refresh all your key material. For example, the disclosure of a private key of a Bitcoin wallet gives access to the coins inside. This makes it unattractive to store a complete backup of your private key(s) with your bank or your spouse. The better option would be to split the key into multiple parts. The recommended way to do this securely is to use the Shamir secret sharing scheme. This talk provides a detailed breakdown of how the scheme works and explains how it is implemented in C in a new library called SSS.
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