This is possible because the computers in question are powered via large batteries hooked to stationary bikes; one minute of pedal power is good for five minutes of computer usage. The five computers use WiFi to transmit to a central location which has a dish antenna. From there, a microwave transmitter connects them to a local hospital "which has two of the region's few phone lines."
As low-tech as bike-powered computers sound, the article says that other communities have used cow manure to power their systems, so this is probably a step up. The system uses open source software and "discarded microchips," and according to the article, the inventor of the first portable computer custom-built the PCs to withstand the harsh climate.
Not bad, especially when you consider that the entire thing (five computers, stationary bikes, batteries, software and network connectivity) was designed and built for $19,000 and costs only $21 a month to operate. Impressive stuff.