Every Federation starship requires amble supplies of deuterium to fuel their propulsion systems. Current deuterium storage protocols call for tanks constructed of forced-matrix 2378 cortanuim and stainless steel, insulated by foamed vac-whisker silicon-copper-duranite. Typically, a vessel will have a primary deuterium tank (PDT) from which fuel is delivered to the warp- and impulse propulsion systems, well as the Reaction Control System. Deuterium in the PDT is stored in slush form (semi-frozen to 13.8Ks), both for safety reasons and because it is the preferred form that is fed into the warp propulsion system.
For impulse and reaction control systems, ships have secondary or auxiliary cryo tanks in which deuterium is stored in liquid form. Transfer between tanks is the norm, with the fuel passing through specialized heating/cooling coils.
No matter what the form, deuterium tanks are given their own set of internal shielding and inertial dampening fields. This, along with the stength of the tank, ensure that the fuel doesn't leak, shift, slosh, or come into contact with items that can pose a threat to the ship and crew. Storage tanks also include fill and vent ports, distribution piping, and sensors. (Star Trek: The Next Generation Technical Guide)
Modern shipbuilding design incorporates tankage that can carry enough deuterium to power a starship for three years. Refueling can be done at most space stations and fleet yards or by rendezvous with a tanker. Refueling, tank monitoring, and storage maintenance are part of the Engineering Department's responsibilities.