Ohne Zirkel kann sie die Aufgabe nicht machen.

Because ohne Zirkel is a prepositional phrase, and German often puts this kind of phrase first for emphasis or style.

The normal basic order would also be:

Sie kann die Aufgabe ohne Zirkel nicht machen.

When ohne Zirkel is moved to the front, the sentence still follows the German verb-second rule, so kann must come immediately after it:

• Ohne Zirkel
  • kann
    • sie
      • ...

So the fronted phrase is not unusual; it just gives a little more focus to the idea of without a compass.

German main clauses usually follow the V2 rule: the finite verb must be in the second position.

In this sentence, the first position is taken by Ohne Zirkel, so the finite verb kann has to come next:

• Ohne Zirkel = position 1
• kann = position 2
• sie = then the subject comes after that

This is very common in German. The subject does not always come first.

Because kann is a modal verb. With modal verbs in German, the main verb usually goes to the end in the infinitive form.

So:

• kann = finite modal verb
• machen = infinitive main verb at the end

This pattern is standard:

• Sie kann schwimmen.
• Er muss arbeiten.
• Wir wollen gehen.

So machen is at the end because it depends on kann.

After ohne, German often leaves out the article when speaking in a general sense, especially when the noun is being treated as an unspecified tool or thing.

So ohne Zirkel means something like:

• without a compass
• not using a compass

You could also say:

Ohne einen Zirkel kann sie die Aufgabe nicht machen.

That is also correct, but it sounds a bit more explicit: without a compass / without one compass.

A rough difference:

• ohne Zirkel = more general
• ohne einen Zirkel = more concrete, more clearly referring to an individual object

Ohne takes the accusative case.

So if there is an article or adjective, it must be accusative:

• ohne einen Zirkel
• ohne den Zirkel
• ohne einen guten Zirkel

In your sentence, Zirkel has no visible article, so you do not see a case ending. But grammatically, it is still governed by ohne, so it is understood as accusative.

Here Zirkel means a drawing compass used in geometry for making circles.

That is important because English speakers may confuse it with circle, but Zirkel does not mean the shape itself. The shape is:

• der Kreis = circle

So:

• der Zirkel = compass
• der Kreis = circle

In this sentence, nicht negates the verbal idea: she is not able to do the task.

With modal verbs, nicht often appears before the final infinitive:

• Sie kann die Aufgabe nicht machen.

This is the normal placement when the whole action is being negated.

Compare:

• Sie kann die Aufgabe nicht machen. = She cannot do the task.
• Sie kann nicht die Aufgabe machen. = This would sound marked and would suggest contrast, as if she can do something else, but not the task.

So the position of nicht helps show what is being negated.

Because die Aufgabe refers to a specific task/exercise.

German often uses the definite article where English may or may not use one, especially when the speaker and listener know which task is meant.

So die Aufgabe suggests:

• the task
• the exercise
• the problem at hand

If you said eine Aufgabe, that would mean a task / one task, not a specific known one.

Yes, that is also correct.

Both are grammatical:

• Ohne Zirkel kann sie die Aufgabe nicht machen.
• Sie kann die Aufgabe ohne Zirkel nicht machen.

The difference is mainly focus and style.

• Ohne Zirkel ... puts early emphasis on the missing tool.
• Sie kann ... starts more neutrally with the subject.

German word order is flexible in this way, as long as the finite verb stays in second position in a main clause.

Because the first position is already occupied by Ohne Zirkel.

In German main clauses, only one element comes before the finite verb. If something other than the subject is placed first, the subject usually moves after the verb:

• Sie kann die Aufgabe nicht machen.
• Ohne Zirkel kann sie die Aufgabe nicht machen.

So sie is after kann simply because German keeps the finite verb in second position.

Yes. Ohne den Zirkel would mean without the compass, referring to a specific compass.

Compare:

• ohne Zirkel = without a compass / without using a compass
• ohne einen Zirkel = without a compass, one compass, any compass
• ohne den Zirkel = without the specific compass

So the choice depends on whether the speaker means a tool in general or a particular known object.

Yes, it is natural. Eine Aufgabe machen or die Aufgabe machen is a common way to say do a task/exercise.

Depending on context, German might also use other verbs, for example:

• die Aufgabe lösen = solve the problem/exercise
• die Aufgabe bearbeiten = work on the task

But die Aufgabe machen is perfectly normal, especially in everyday language and in school contexts.

Here kann most naturally expresses ability/practical possibility.

It means that without the compass, she is not able to do the task, or it is not possible for her to do it properly.

It does not normally suggest permission here. If the meaning were permission, German would more often use:

• darf = may / is allowed to

So in this sentence, kann is best understood as can / is able to.