Срок аренды на складе индивидуального хранения уже закончился.

Срок means a term, period, or deadline with a defined limit. In this sentence, it refers to the rental period.

Время is a much broader word meaning time in general.
So:

• срок аренды = the rental term / rental period
• время аренды would sound less precise here

Russian often uses срок when something has an official or fixed duration.

Because аренды is in the genitive case.

The phrase срок аренды literally means the term of the rental. In Russian, nouns often go into the genitive after another noun to show this kind of relationship.

Compare:

• срок аренды = rental term
• стоимость аренды = cost of the rental
• начало аренды = beginning of the rental

So аренды is not the subject here; it is dependent on срок.

Russian normally says на складе for at the warehouse / in the storage facility.

Even though English would often say in, Russian uses на with some places and institutions where English learners might expect в. This is just the usual idiomatic choice.

So:

• на складе = at/in the warehouse
• на почте = at the post office
• на заводе = at the factory

Using в складе would sound unnatural in this meaning.

This refers to a self-storage facility or a storage facility for individual use.

Literally:

• склад = warehouse / storage facility
• индивидуального хранения = of individual storage

So the phrase is somewhat more formal and descriptive than the usual English term self-storage.

A very literal rendering would be something like at a warehouse for individual storage, but in natural English that is usually just at the self-storage facility.

Because индивидуального хранения is a genitive phrase modifying складе.

Here is the structure:

• на складе = at the warehouse / in the storage facility
• индивидуального хранения = of individual storage

And inside that phrase:

• хранения is genitive singular of хранение
• индивидуального agrees with хранения in case, number, and gender

So индивидуального хранения works as one unit describing what kind of warehouse it is.

Because it agrees with the subject срок, which is a masculine singular noun.

In the past tense, Russian verbs show gender and number:

• masculine: закончился
• feminine: закончилась
• neuter: закончилось
• plural: закончились

Since the subject is срок, you get:

• Срок ... закончился.

If the subject were аренда, it would be:

• Аренда закончилась.

Because закончился is perfective, and it means the action is completed: the rental term has ended / ended.

Russian aspect matters here:

• закончился = ended, came to an end
• заканчивался = was ending, used to end, was coming to an end

In this sentence, the speaker is stating a completed fact, so perfective is the natural choice.

Уже means already.

It emphasizes that the rental term has ended by now, perhaps earlier than expected or as a relevant fact at the current moment.

Compare:

• Срок аренды закончился. = The rental term ended.
• Срок аренды уже закончился. = The rental term has already ended.

So уже adds a sense of already / by this point.

Yes. Срок аренды истёк is also very natural and often sounds a bit more formal or official.

Compare:

• Срок аренды закончился = the rental term ended
• Срок аренды истёк = the rental term expired

Both are correct.
Истёк is especially common with contracts, deadlines, terms, and official periods.

Yes, Russian word order is fairly flexible.

The neutral order here is:

• Срок аренды на складе индивидуального хранения уже закончился.

This is natural and clear: subject first, then details, then the verb.

But you could also move things for emphasis, for example:

• На складе индивидуального хранения срок аренды уже закончился.
  This emphasizes the location first.

Russian changes word order more freely than English, but not every order sounds equally neutral. The original sentence is a normal, standard way to say it.

In meaning, it tells you where this rental term applies: at the self-storage facility.

So semantically it is closely tied to аренды:

• срок аренды [where?] на складе индивидуального хранения

But in the sentence as a whole, it functions as a location-related modifier for the situation being described.

A natural way to understand it is:

• The rental term for a unit at the self-storage facility has already ended.

So it is not saying the term itself is physically located somewhere; it is saying the rental arrangement concerns that place.

Yes, a Russian speaker can say this, and it is understandable. But depending on context, other phrasings may sound more natural.

Possible alternatives include:

• Срок аренды склада индивидуального хранения уже закончился.
• Срок аренды бокса на складе индивидуального хранения уже закончился.
• Срок аренды ячейки уже закончился.
  if the exact type of storage unit is already known

Sometimes Russian becomes more natural when it names the thing being rented more directly, such as:

• бокс = storage unit
• ячейка = locker/unit
• место = space

So the original sentence is fine, but in real usage the exact noun may vary depending on what is being rented.

You can translate it literally enough to understand the structure:

• Срок = term
• аренды = of the rental
• на складе = at the warehouse / storage facility
• индивидуального хранения = of individual storage
• уже = already
• закончился = ended

So the literal sense is:

The rental term at the individual-storage warehouse has already ended.

But in natural English, you would usually say something more like:

The rental period at the self-storage facility has already ended.

So the literal translation helps with grammar, but the natural translation sounds smoother.