The scenario, if I may summarize it from the OP in my own words:

 - a closed system (the spaceship) that needs to remain closed for 6 months.
 - 7 people inside that need to eat to stay alive for that period.
 - some food cycle system did exist in the ship, but some part is now broken. 
 - thus a severely diminished food supply that will not sustain the lives of the 7 for that duration - but not completely zero.

I'm also going to assume that:

 - no resupply from the target planet was possible, either through lifelessness or incompatibility of native organic matter with the Earth-originated ecology.
 - the spaceship can't make any intermediate stops for resupply either (myriad of reasons possible).
 - the ship has some form of artificial gravity, as that will make some of the processes much easier and a known quantity (already done on Earth).

What I propose is alluded to in some other answers and comments, but I'm going to flesh it out in practical terms:

Do nutrient recycling as it is done on Earth by natural processes
===
(enhanced by human intervention, as needed).

 1. **Time:** What people often underestimate regarding growing plant crops is the time that is needed. It's not simple sticking some seeds in soil and watering them, and *voilà* the next day (or month) a fully-developed, nutritious potato has appeared. Same goes for breaking down biomass until it is suitable as a growing medium that provides nutrition to the plants that grow in it.  You need to provide some means for the initially needed time to ramp up the system - perhaps by having some remaining foodstuffs, strictly rationed; perhaps by having the crew fast (30-40 days is probably realistic without adverse long-term health effects).

     a) I'm not sure how you will get *very hungry* humans to abstain from "touching" the not fully grown plants. The crop needs time to develop to full potential where it can feed the entire crew. If I were to write such a story, maybe I'd have the expert/hobby botanist on the crew locking/barricading him/herself into the agri section (with some rations) were s/he gets the process going, only coming out once food supply has been guaranteed. At least there is enough plot potential for an entire book right there - tough love, hard choices between two suboptimal options, isolation, conflict, power dynamics and politics, human nature, and all that.

 2. Converting biomass to growing medium can be accomplished in a myriad of ways.

    a) **Lasagna containers:** People I know build "lasagna beds" in plastic tote cases, old truck tires, or similar containers, which can be done in small courtyards or even in sunny indoor places. Recommended depth around 40 cm / 16 inch, and would be done in a continuous fashion as biomass becomes available and new crops need to be established. Read up on the lasagna method, but it entails alternating layers of "green" (nitrogen rich) and "brown" (carbon rich) biomass, which triggers the composting process. Paper and cardboard generally count under "browns". Can be enhanced with some soil, compost (from previous batches), minerals. Can be planted immediately if the top layer is an appropriate compost/soil mixture, by the time the roots have reached the lower levels, they should be well underway to compost state. Needs sufficient space for the composting process to develop (depth x width x length) so a tote-size case is probably minimum per iteration.

    b) One can also make **compost** outright. The speediest compost piles are "warm piles", i.e. microbe-rich (more on that later) which are given optimal circumstances to do the decomposition at optimal rate and generate quite some heat in the process. You'd need a space of dimensions about 1 m / 3.5 ft min to 1.5 m / 5 ft max, to contain the composting material, and some means to ventilate it (blowing air through it mechanically should work). "Dimension" means length, width, height and/or diameter, as applicable - it's a flexible guideline. Again "greens" and "browns", mixed well, in a ratio of around 1:1 to 1:2, and quite a lot of moisture (water) - wet but not soggy or dripping. One or two "turn-overs" (remixing the pile) are often needed to keep the decomposition going, so you'd (temporarily) need that extra space to keep the material. Let's say you can have growing medium ready in about 6 weeks. Composters are usually discouraged from putting anything other than plant material in it, but "warm" compost piles as above have been known to decompose various dairy products, meat (raw and cooked), cooked leftovers, oily foods, hair, bones, blood, and even (small) animal carcasses and human waste products (urine and excrement). Warm piles have been known to ignite in rare cases, which is usually prevented by not going too big (or monitoring them like a baby).

    c) Another way that is used in many kitchens (including small apartments) to handle biomass is the **Bokashi** method. This can also handle all of the above non-plant materials. A filled container, mixed with the bokashi "bran", is closed air-tight for 2 weeks to allow an anaerobic fermentation process to complete, which is performed by various microbes with which the "bran" was inoculated. Do note however that this is not a complete process, and would require finishing-off by one of the other methods - the benefit however is that it preprocessed some biomass that the other methods would not like, or take much longer with, so the recycling is sped up.

    d) **Earthworms** are marvelous little creatures that in 24 hours can turn biomass of half their body weight to the absolute best compost (often wonderingly called "black gold"). However, they work best on half-rotted material (compost pile or bokashi thus works well as a preprocess step).

 3. Once you've got your biomass nutrients turned into growth medium in this way, I think you're past the biggest obstacle since this nutrient and microbe-rich material grows plants that are usually disease free and producing abundantly (provided the right minerals and trace elements are present). Since it is so nutrient-rich, it lends itself to higher plant densities or smaller growing containers, which may be appropriate in a space where space is at a premium. Of course, this assumes that the necessary seeds/seedlings/cloneable plant material is on board in the first place...

    a) The earthworms also produce "castings", a mass of little roundish particles that some have converted hydroponics systems to use instead of the usual growing salts - see [vermiponics](https://en.wikipedia.org/wiki/Vermiponics). Otherwise they could just be added to the growing medium. You would probably still get such medium out of your worm farms (and/or other systems) in addition to the castings.

 4. Apart from eating plants, I have heard earthworms are also considered a good source of complete proteins and a delicacy in some places - prepared in the right way. (No first-hand knowledge here, (un)fortunately...) Guesstimates are that you can keep not more than 5 kg of Eisenia fetida worms per 1 m² (about 1 pound per sq ft). You'll have to research reproduction rate yourself if you want to go this route, but they can be quite good breeders under the right circumstances.

Some problems and other notes
===
 1. The above processes do depend on microbes (viruses, bacteria, yeasts; and optionally but preferably also nematodes, worms, insects) to work *at all*. They occur on Earth simply in the environment, but what about on a spaceship that's presumably sterilized? Although, the International Space Station has [shown that it is quite hard to keep all microbes out](https://www.forbes.com/sites/linhanhcat/2019/04/17/microbes-international-space-station/). All living organisms on Earth live in symbiosis and we humans probably carry more microbes than our own cells around with us. So some mix of luck, microbiology knowledge, accidentally or deliberately brought-along microbiomes would be needed to get the right critters for the process. Maybe have bokashi be a part of the original waste conversion process, some on-board experiments, a smuggled-in pot plant, fecal swabs of the whole crew by the microbiologist....

 2. Same goes for earthworms. Why would there be [Eisenia fetida](https://en.wikipedia.org/wiki/Eisenia_fetida) on board? Some reason to be thought of.

 3. Human waste products (urine and excrement) are a nutrient resource that should not go to waste in a closed system like this spaceship. Warm compost piles and bokashi fermentation both have the potential to neutralize harmful pathogens and help return the material to the food chain. It has already been done on Earth. If you need more background, I strongly recommend [The Humanure Handbook](http://humanurehandbook.com/) as a starting point, in addition to websearch for the right terms. This book also provides a wealth of information about warm compost piles and microbial life, even if one does not plan to bring any animal dung near it.