A food system would be as simple as merely having food and the player character eating it automatically. Keeping it as a purely survivalistic aspect passes up a prime roleplaying opportunity, however.
Say, for instance, the Pipboy has a mealplanner. In this mealplanner, the player can set the player character's diet based on what foods are available to the party. The kinds of foods the player character eats may have certain positive and negative side effects.
Breads for instance, wouldn't have very significant positive or negative effects, and something like the limitless shelf-life of pre-war calorie rations would have no effects.
A diet high in brahmin meat or jerky, however, would be high in protein. This would lead to a slight increase for unarmed and melee combat modifiers, and the eventual acquisition of a strength boost. The player character would have to maintain this high protein diet to maintain the stat boost, though, in order to keep the player from abusing food related perks.
Other diets could lead to the acquisition of traits as well. A diet high in gecko meat, for instance, could lead to the acquisition of the Gecko Eater trait. A player character with Gecko Eater would have a higher rad resistance, but at the same time would suffer negative effects on its sex appeal, which would manifest by NPC's referring to it as "gecko breath."
A cooking skill has potential, but making it for the preservation of food puts too much strain on the player in regards to experience allocation.
Instead, a cooking skill could be dedicated to the preparation of food in the post-apocalyptic world. A high cooking skill would lead to better positive diet benefits and less significant negative side affects. Higher cooking skills would also lead to a faster acquistion of perks, such as the beforementioned strenght boost, and traits like the Gecko Eater.
A high cooking skill would also allow the player character to participate effectively in cooking related side quests. Say, for instance, an agrarian community has sprung up in a more fertile area of the wastes, such as Modoc and the Ghost Farm. Every year, the community would celebrate a harvest festival that lasts for a week. This festival would be implemented by a more "festive" town appearance and the ability to participate in an Iron Chef style contest. The precept being that it was inspired by pre-war pop culture. A vault dweller player character with a high perception would recognize this from his or her viewing of old entertainment reels in the vault. The participants would be given a "secret ingredient" to work with, and the player character's cooking skill would determine what dishes the player could select to make in dialogue options. To keep this from being too frustrating, all the dishes would have the same effect since they're all in the same quality bracket determined by the player character's skill. The joke behind the festival would be that if the player character returned for another year's festival, the secret ingredient would be the same thing. This ingredient would also be the community's chief product.
A refridgeration unit could also be acquired for the player character's vehicle. The benefits being that it'd take longer for food to spoil. The drawbacks, though, would be less space in the trunk, and a greater energy demand from the vehicle.
The player character could recruit a "Cookie" PNPC. The benefits would be enjoying the aspect of a high cooking skill without having to invest in the cooking skill itself, but Cookie would be slightly sub-par in combat, and the player character couldn't participate effectively in cooking related side quests.
In order to unreqruit an NPC, the player character would have to give him or her some food to convince them to leave. The amount of food it would take to convince them to leave would depend on the location you unreqruit them. In a town, the NPC's demand for food would be less, but out in the middle of the wastes, the player can expect to receive a significant demand.