I've had a few power cuts on an ubuntu box running bitcoind which requires me to re-index the blockchain. Is it possible to shut down bitcoind each night, backup the chain state and block height, then restart bitcoind? Then if a power failure occurs, I can just restore last good chain state?
Potentially, yes; look into using rsync or a filesystem with checkpoints for your backups. However, depending on the severity and duration of the power cuts, it may be easier to buy a UPS to allow your ubuntu box to keep running through a power failure (or failing that, at least provide it with plenty of time to shut down safely).