The Interior of Iceland is a rugged snowy territory, accessible only in summer.
None, actually! The interior is vast but has no permanent population. The closest thing it has to settlements of any kind are huts, some of which are manned during the summertime, providing accomodation for travellers.
Anyone's biggest worry in the interior is to run out of petrol. Here are a few places to look out for. Accommodation is available in all of them as well as food.
The easiest, cheapest and safest way to venture into the interior is by BSI bus (special 4x4 buses with experienced drivers). Renting a 4X4 is very expensive and must be done in advance. It's best to travel with at least one other car as conditions are extreme. Remember that off-road driving is prohibited in Iceland because it contributes to erosion and can be very dangerous.
Bring your own provisions. There is no guarantee that you will find food anywhere.
You can safely drink water from any stream.
Every year, people die in the interior due to being caught by bad weather and being unprepared.
Do not venture into the interior if you don't know what you're doing, or are not accompanied by somebody who knows what they're doing. It is a region of extreme weather with no settlements, so you need to be prepared for whatever may happen. This includes packing very warm clothes and bringing food and drinks for a stay that may become longer than inteded. You must also have a good car, and for most roads there 4x4 trucks are neccesary. Never travel into the interior without being sure somebody at home knows your itinerary and can contact the authorities if they don't hear from you within a set time.