Does Bitcoin rely on the internet? Is it possible to run Bitcoin mining and transaction distribution without using the internet? 

For example, how would Bitcoin continue in the event of a total clampdown by a telco?
You can technically run appropriately modified Bitcoin on any sort of network of communication, be it LAN, mesh, or otherwise. One could imagine making a mesh network on Zigbee or Bluetooth and allow anyone to connect to those without the Internet, although then the network latency would be way bigger and the Bitcoin network could easily become forked. Preferably there would be at least one proxy server running both on the internet (be it through wires or cellular) and on the distributed network.

If you want to make payments without using the internet, use physical coins, simple as that. It is far safer than making a new internet-less network. The only way you can lose in this way is if the Bitcoin value would plummet.