How is fuel stored in a tanker aircraft?
You might be surprised to learn that most refueling tankers hold a relatively small percentage of their total fuselage volume in fuel. The reason why is simple; a plane stuffed to the gills with fuel would be so grossly overloaded it wouldn't make it off the runway. Tankers, therefore, can double as medium transports with little modification, as they have a sizable upper deck which is more or less empty space when serving as a tanker, but which can ferry troops, light vehicles or other materiel from Stateside bases to various theaters (a dedicated heavy transport like the C-5 or C-17 would be used for transporting large units or for specialized troop transport like airbornes).

Here's the layout of the KC-135 Stratotanker:



To my knowledge, all of this fuel is available both to power the plane and to refuel other aircraft, however the plane would likely draw primarily from the wing tanks for flight, as this is the "standard" configuration for the 707 the KC-135 is built on. Transport of the fuel from these tanks to the refueling systems would be accomplished with an integrated pump and pipe system that could move fuel from anywhere to anywhere.

The bulk of the fuel load, as you can see, is contained in four "body tanks" totaling over 144,000 pounds of fuel. For comparison, the F-15E strike fighter can carry a maximum, with CFTs and 3 droptanks, of just 35,550 pounds. 

These tanks are lined with plastic bladders, which contain the fuel (so the entire fuselage doesn't have to be leak-proof which is much more difficult and expensive to ensure) and also allows the bladder to collapse as the fuel inside it is depleted, reducing the volume of air in direct contact with the fuel, thus reducing the vapor hazard. AFAIK the wing tanks do not have these bladders, but they're much smaller spaces overall and more easily accessed. 

Separating the fuel into multiple tanks has several advantages:


The tanker is still serviceable with one or more of the tanks unable to carry fuel (leak in the bladder, pump failure, etc).
Sloshing of the fuel in the tanks has less effect on the weight distribution of the aircraft. Imagine trying to take off with a full Olympic swimming pool in your cargo hold; any pitch up or down will cause several tons of weight to move "downslope", unbalancing the aircraft (probably fatally).
Fuel can be pumped between these tanks to fine-tune or "trim" the aircraft's weight distribution and handling characteristics, a process known as "tank trimming" which is more aerodynamically efficient than elevator trimming.