I've often thought  a temporary cell phone base station for emergency situations would be useful but I wasn't sure whether they already exist. Also I don't know how it would connect to the 'outside world' or be powered. Have you already solved these problems?

My usage scenario is say after a hurricane or  earthquake  eg Haiti where people need to contact their loved ones and find online lists of those missing or found, and safe places to go.

Some definitely do exist, but the ones we've found are quite expensive and not always easy to deploy. Your usage scenario is very much what we envision as well. In order to connect it to the "outside world," we're taking advantage of the amateur radio-based Automatic Position and Reporting System (APRS). This was an old FEMA system developed in the 90s that amateur radio operators use when coordinating disaster relief.  In our system, if someone connects to the tower using their Android phone, it reports their status (OK/not OK), their position, and allows anyone on the APRS network to send information direct to the user. Combining this with a WiFi mesh network, we're hoping this adds a higher degree of visibility to anyone who is in direct need of assistance as well as allow users to be in direct contact with their loved ones in the event they're separated. 

The power supply is a 12V system. There's an internal battery, as well as a way to connect the battery to a wall-charger or additional 12V supply (e.g. car battery, solar panel).