Sem a fúrás, sem a kopogás nem zavaró.

Sem ... sem is the standard Hungarian pattern for neither ... nor.

So:

• sem a fúrás = neither the drilling
• sem a kopogás = nor the knocking

Together, Sem a fúrás, sem a kopogás ... sets up two things that are both being denied.

This pattern can be extended to more than two items:

• Sem a zaj, sem a fény, sem a szag nem kellemes.

Yes, but Hungarian normally still uses nem with the predicate.

So the structure is:

• Sem X, sem Y nem ...

This is normal Hungarian grammar, not a mistake. English usually uses just neither ... nor without another negative word, but Hungarian commonly keeps the negative marking on the verb or predicate too.

So:

• Sem Péter, sem Anna nem jön.
• Sem a fúrás, sem a kopogás nem zavaró.

A good way to remember it is: sem ... sem introduces the negative subject(s), and nem negates the predicate.

Because both nouns are being used with the definite article: a fúrás, a kopogás.

In a sem ... sem construction, each coordinated noun phrase usually keeps its own article:

• Sem a tanár, sem a diák nem késett.

So it is natural to repeat the article:

• sem a fúrás, sem a kopogás

This is similar to English neither the drilling nor the knocking.

Hungarian usually omits the verb van (to be) in the present tense, third person, when the predicate is a noun or adjective.

So:

• A zaj zavaró. = The noise is disturbing.
• A zaj nem zavaró. = The noise is not disturbing.

That is why nem zavaró already means is not disturbing.

But in other tenses, or in 1st/2nd person, the verb appears:

• A zaj nem volt zavaró. = The noise was not disturbing.
• Nem vagyok fáradt. = I am not tired.

With sem ... sem, Hungarian very often uses a singular predicate, even when two items are mentioned.

So:

• Sem Péter, sem Anna nem magas.
• Sem a fúrás, sem a kopogás nem zavaró.

The idea is a bit like neither X nor Y is ... rather than X and Y are ....

If you used és (and) instead, a plural predicate would be more likely:

• A fúrás és a kopogás zavaróak. = The drilling and the knocking are disturbing.

So the singular zavaró is normal here.

They are verbal nouns or action nouns.

They come from verbs:

• fúr = to drill → fúrás = drilling
• kopog = to knock / tap → kopogás = knocking / tapping

The suffix -ás / -és is very common for making nouns from verbs:

• olvas → olvasás = reading
• tanul → tanulás = studying
• fizet → fizetés = payment / paying

In this sentence, fúrás and kopogás behave just like ordinary nouns.

Zavaró is an adjectival form built from the verb zavar (to disturb, bother, disrupt).

The ending -ó / -ő often creates a kind of participial adjective:

• zavar → zavaró = disturbing
• érdekel → érdekes is different, so not every adjective follows this pattern, but -ó/-ő is still very common in verb-based adjectives
• fáraszt → fárasztó = tiring

In this sentence, zavaró functions as an adjective meaning disturbing / bothersome.

Yes, Hungarian word order is flexible, but not random.

The given sentence is a very natural, straightforward version:

• Sem a fúrás, sem a kopogás nem zavaró.

You could also say:

• Nem zavaró sem a fúrás, sem a kopogás.

That version puts more emphasis on not disturbing.

What usually stays important is this pattern:

• sem ... sem
• nem directly before the predicate

So even when the order shifts, nem zavaró stays together.

Grammatically, it names the actions or events: the drilling and the knocking.

But in real use, Hungarian often uses these action nouns to refer loosely to the associated noise or disturbance too. So depending on context, a fúrás can mean:

• the act of drilling
• the drilling noise
• the drilling going on nearby

That is very natural in Hungarian, just as in English the drilling can refer to both the action and the sound.