At the core of Bitcoin technology is the blockchain, the public ledger that lists every transaction that has ever happened. The blockchain is typically transmitted through the internet using the p2p technology built into the Bitcoin Core client. Nodes broadcast their copy of the blockchain and it is constantly confirmed and updated between each node. When miner's solve a new block, they then broadcast the new block to the nodes and it is propagated throughout the network. As it stands now, without internet access, a person's ability to use Bitcoin is greatly reduced, specifically accepting Bitcoin as a payment becomes impossible because you cannot check to see if a transaction has been confirmed in the blockchain without internet access. Fortunately, there are ways to mitigate this issue.
Sending Bitcoin in situations with no internet can be solved through the use of a 3rd party sms wallet provider. Companies like 37 Coins are making this a reality today. Theoretically, you could also use morse code to send transactions over traditional radio communications but a wallet provider would need to emerge and offer that service.
Receiving the blockchain without internet access may soon be a possibility. Long time Bitcoin Core Developer, Jeff Garzik, announced plans earlier this year to launch a cube satellite that will broadcast the blockchain worldwide from space. Alternatively, there is now a Finnish project under development with the goal of broadcasting blockchain data and current exchange price over DVB-T. Theoretically, someone else could do a similar project using a different radio standard, such as FM or amateur radio frequencies. The project is currently taking Bitcoin donations, but make sure you send any donations from addresses where you control the private keys because if they don't reach their funding goal they will refund your Bitcoin back to the address it came from.
Last but not least, as Bitcoin users we have to help support the development and roll-out of cheap and reliable internet access on a global scale. We also need to support the free flow of information, because if the internet is allowed to be censored by governments and/or corporations, the effect on open source development could be chilling.