Is Bitcoin inherently dependent on electricity? Obviously, the computing power necessary to create the blockchain requires access to fast hashing which requires electricity. Given that local power outages do occur, and this can cause a lack of network availability, does this act as the Achilles heel for Bitcoin?

Is it possible to conduct Bitcoin transactions in the absence of electricity?
The network is distributed, so it won't go down everywhere at once.

Locally, on one hand it is similar to debit transactions - they need to get online to be processed. One can use cellphones though, which have their own batteries to run for awhile, so it's not too bad.

Lastly, one can accept transactions offline (using QR codes for example) and later transmit them through the network, but that carries the risk of double spending.

And if you want truly offline transactions with little risk, use things like Casascius physical Bitcoins.