Onder deze garantie volgt de terugbetaling binnen vijf dagen.

Because Dutch is a verb-second (V2) language in main clauses.

The sentence starts with Onder deze garantie, so that counts as the first element. In a Dutch main clause, the finite verb must then come second:

• Onder deze garantie
  • volgt
    • de terugbetaling
      • binnen vijf dagen

If you started with the subject instead, you would get:

• De terugbetaling volgt binnen vijf dagen onder deze garantie.

That is also grammatical, but the original sentence puts extra focus on the condition onder deze garantie.

Here onder is not mainly physical. It means something like:

• under this guarantee
• according to this guarantee
• covered by this guarantee

Dutch uses onder in this abstract sense quite often. So onder deze garantie means that the repayment happens within the terms of this guarantee or if this guarantee applies.

Literally, volgen often means to follow. In a formal sentence like this, volgt means something like:

• takes place
• comes
• will occur
• follows

So de terugbetaling volgt binnen vijf dagen means that the repayment will happen within five days.

This use is a bit formal or business-like. In more everyday Dutch, people might also say:

• De terugbetaling gebeurt binnen vijf dagen.
• U krijgt binnen vijf dagen uw geld terug.
• Het bedrag wordt binnen vijf dagen terugbetaald.

Dutch often uses the present tense to talk about the future when the context already makes the timing clear.

Here, binnen vijf dagen clearly points to the future, so volgt is enough. Dutch does not need a separate future form as often as English does.

So this is normal Dutch:

• De terugbetaling volgt binnen vijf dagen.

A more explicitly future version is also possible:

• De terugbetaling zal binnen vijf dagen volgen.

But the version without zal is very common and natural, especially in formal statements.

Because terugbetaling is a noun here, and Dutch usually needs an article when referring to a specific, identifiable thing.

• de terugbetaling = the repayment / the refund

Without the article, it would sound incomplete in this sentence.

Also, terugbetaling is a de-word noun, so the article is de, not het.

It is a compound noun:

• terug = back
• betaling = payment

So terugbetaling literally means paying back, and in normal English that becomes:

• repayment
• refund

Dutch forms many nouns this way by combining smaller parts into one word.

It means within five days, so no later than five days from now / from the relevant starting point.

It does not mean the action lasts for five days. It means the action will happen at some point before the five-day limit is reached.

So:

• binnen vijf dagen = sometime during the next five days, at the latest by day five

Because this is a standard Dutch time expression.

After binnen when giving a deadline or time limit, Dutch normally says:

• binnen een uur
• binnen twee weken
• binnen vijf dagen

So vijf dagen here is just the bare plural noun phrase, like English within five days.

Adding an article, such as binnen de vijf dagen, would usually sound unnatural here.

This is a very useful distinction.

• binnen vijf dagen = within five days, before the deadline of five days
• in vijf dagen = in five days or sometimes over the course of five days, depending on context

In your sentence, binnen vijf dagen is the correct choice because it gives a maximum time limit.

Examples:

• U ontvangt een reactie binnen vijf dagen.
  = You will receive a response within five days.

• Ik heb het rapport in vijf dagen geschreven.
  = I wrote the report in five days.

So binnen is about a deadline; in is often about duration or a point in time.

Yes, that is grammatically possible.

Dutch allows different word orders depending on what you want to emphasize. Compare:

• Onder deze garantie volgt de terugbetaling binnen vijf dagen.
• De terugbetaling volgt binnen vijf dagen onder deze garantie.

The original sentence sounds more formal and gives early emphasis to the condition onder deze garantie. That is often useful in legal, administrative, or customer-service language.

So the original order is not random; it helps frame the whole statement as applying under this guarantee.

Yes, it sounds fairly formal and written.

A native speaker would understand it easily, but in everyday conversation people might choose something more direct, such as:

• Onder deze garantie krijgt u binnen vijf dagen uw geld terug.
• We betalen binnen vijf dagen terug onder deze garantie.
• Het bedrag wordt binnen vijf dagen terugbetaald.

The original sentence is the kind of wording you might see in:

• terms and conditions
• a warranty policy
• a company website
• customer service documentation

Because garantie is a de-word.

In Dutch:

• with de-words, you use deze
• with het-words, you use dit

So:

• de garantie → deze garantie
• het boek → dit boek

That is why deze garantie is correct.