Вес этой коробки небольшой, зато объём у неё такой, что она занимает половину багажника.

Because Russian often expresses the X of something with a noun in the genitive case.

• вес этой коробки = the weight of this box
• цвет машины = the color of the car
• размер комнаты = the size of the room

So коробки is genitive singular of коробка, and этой agrees with it in case, gender, and number.

Because it agrees with вес, not with коробки.

The subject here is вес (weight), and вес is a masculine singular noun. So the predicate adjective is also masculine singular:

• Вес ... небольшой
• Объём ... большой
• Размер ... маленький

Even though коробка is feminine, the sentence is describing the weight, not the box directly.

Yes, basically. Небольшой literally means not big, but in many contexts it simply means small or rather small/modest.

Here it sounds a little more neutral and natural than a blunt маленький:

• вес небольшой = the weight is not great / is fairly small
• маленький вес is possible, but less standard in this kind of statement

Russian often prefers небольшой with things like weight, size, amount, distance, etc.

Зато introduces a contrast, often like but on the other hand, however, or the trade-off is that.

In this sentence, the idea is:

• the box is not heavy
• but its volume is very large

So зато is not just plain but. It often suggests a balancing contrast:

• Далеко, зато красиво. = It’s far, but at least it’s beautiful.
• Недорого, зато долго ждать. = It’s inexpensive, but the downside is a long wait.

Here: the weight is small, but the volume is a different story.

Both are possible, but объём у неё is a very common Russian way to say as for it, its volume... or it has a volume...

The pattern у + genitive is often used to talk about what someone or something has:

• У неё большой объём. = It has a large volume.
• У машины мощный двигатель. = The car has a powerful engine.

So объём у неё такой sounds natural and conversational.
Её объём is also grammatical, but it can sound a little more formal or more tightly possessive.

Yes, неё is normally her, but in Russian the same pronoun forms are also used for feminine nouns.

Since коробка is feminine, у неё here means for it / of it / it has, referring back to коробка.

So:

• коробка = feminine noun
• у неё = the box has / its...

Russian does this all the time:

• У машины сломалось колесо. = The car’s wheel broke / The car has a broken wheel.
• У книги интересное название. = The book has an interesting title.

This is a very common pattern meaning so ... that ...

• такой, что ... = such that ... / so ... that ...
• такой большой, что... = so big that...
• такой шум, что... = such noise that...

Here, Russian leaves большой implied:

• объём у неё такой, что...
• literally: its volume is such that...
• natural English: its volume is so large that...

So the phrase introduces the result: the box takes up half the trunk.

Russian often omits an adjective if it is obvious from the context.

Here, because the sentence is talking about объём, the listener naturally understands:

• объём у неё такой [большой], что...

Leaving out большой makes the sentence a little more compact and natural. English does something similar sometimes, but Russian does it quite freely in this kind of structure.

It refers to the box.

Grammatically, both коробка and она are feminine, but logically the thing that occupies half the trunk is the box, not the abstract volume.

So the meaning is:

• its volume is so large that the box takes up half the trunk

Russian often allows this kind of shift, where one clause mentions a property and the next clause goes back to the object itself.

Because занимать means to occupy / take up, and it takes a direct object in the accusative.

So:

• занимает половину = takes up half

Then половина governs the genitive:

• половина чего? = half of what?
• половину багажника = half of the trunk

That is why you get:

• половину = accusative, because it is the direct object of занимает
• багажника = genitive, because it depends on половина

Because половина always takes the genitive for the thing being divided.

Examples:

• половина комнаты = half of the room
• половина дня = half of the day
• половину яблока = half of the apple

So in the sentence:

• занимает половину багажника
• literally: occupies half of the trunk

Usually, yes. Багажник most commonly means the trunk of a car in American English, or the boot in British English.

Depending on context, it can also refer to a luggage rack, but here занимает половину багажника clearly means the storage space in a car.

Both commas are required.

1. Comma before зато
   This separates two coordinated clauses:

• Вес этой коробки небольшой
• зато объём у неё такой...

2. Comma before что
   This introduces a subordinate clause:

• такой, что она занимает...
• so ... that it takes up...

Russian punctuation is generally stricter than English with these clause boundaries.

Yes, absolutely. That is also natural Russian.

Compare:

• Вес этой коробки небольшой
• У этой коробки небольшой вес

Both mean essentially the same thing: this box doesn’t weigh much.

The first version sounds a bit more like a neutral statement about the weight as a property.
The second uses the common у + genitive possession pattern and sounds slightly more conversational.

Russian has no articles, so nouns like вес, объём, and багажник appear without a, an, or the.

Also, объём is singular because we are talking about the box’s single overall volume, not multiple volumes.

So English needs something like:

• the volume
• its volume

But Russian simply says:

• объём

The exact English article depends on context, not on anything explicitly present in the Russian sentence.