Jeg er så træt, at jeg næsten ikke kan holde øjnene åbne.

Så ... at is a very common Danish pattern meaning so ... that.

• Jeg er så træt = I am so tired
• at jeg næsten ikke kan ... = that I can almost not ...

So the structure is:

• så + adjective/adverb + at + clause

Examples:

• Han er så høj, at han rammer loftet. = He is so tall that he hits the ceiling.
• Det er så dyrt, at jeg ikke vil købe det. = It is so expensive that I do not want to buy it.

In your sentence, så træt, at ... works exactly like English so tired that ...

Træt is the adjective meaning tired.

After er, Danish usually uses an adjective in its basic form when describing a person:

• Jeg er træt. = I am tired.
• Hun er glad. = She is happy.
• Vi er klar. = We are ready.

Here, jeg is singular, but after er you still use the normal predicate adjective form: træt.

You do not need to change it for person the way some languages do.

Because at jeg næsten ikke kan holde øjnene åbne is a full subordinate clause, and Danish normally needs the subject stated again.

So:

• Jeg er så træt = main clause
• at jeg næsten ikke kan holde øjnene åbne = subordinate clause

Even though English also repeats I here, learners sometimes expect Danish to skip it. It does not. The subordinate clause needs its own subject:

• ... at jeg ...
• ... at han ...
• ... at vi ...

This is standard Danish subordinate clause word order.

In a subordinate clause introduced by at, words like ikke usually come before the finite verb.

Compare:

• Main clause: Jeg kan næsten ikke holde øjnene åbne.
• Subordinate clause: ... at jeg næsten ikke kan holde øjnene åbne.

Notice that in the subordinate clause, kan comes later than an English speaker might first expect.

A useful rule:

• In main clauses, the finite verb is usually in position 2.
• In subordinate clauses, that V2 rule does not apply, and things like ikke usually come before the finite verb.

Næsten ikke means almost not or more naturally in English, can hardly / can barely in this context.

So:

• jeg næsten ikke kan holde øjnene åbne = I can almost not keep my eyes open = more natural English: I can barely keep my eyes open

A subtle point:

• næsten = almost
• ikke = not

Together they soften the negation: it is not completely impossible, but very nearly.

Kan means can / am able to, and holde literally means hold / keep.

In Danish, holde øjnene åbne is the normal expression for keep one’s eyes open.

So:

• kan holde = can keep
• holde øjnene åbne = keep the eyes open

This is very similar to English. It is a natural idiomatic expression, especially when talking about being tired.

Danish often uses the definite form of body parts where English often uses a possessive.

So Danish commonly says:

• holde øjnene åbne = literally keep the eyes open

Even though English often prefers:

• keep my eyes open

This is completely normal in Danish, especially when the owner is obvious from the context.

Other examples:

• Jeg vasker hænderne. = I wash my hands.
• Hun børster tænderne. = She brushes her teeth.
• Han løftede armen. = He lifted his arm.

You can sometimes use a possessive like mine øjne, but here øjnene is the most natural choice.

Here åbne is used in the expression holde noget åbent/åbne, where the adjective agrees with the object in a set way that often looks plural when the object is plural.

Because the object is øjnene (the eyes), Danish uses åbne.

Compare:

• holde døren åben = keep the door open
• holde vinduet åbent = keep the window open
• holde øjnene åbne = keep the eyes open

So the form changes with the noun:

• common gender singular: åben
• neuter singular: åbent
• plural/definite adjective use: åbne

It can be both, depending on context.

In this sentence, it is mostly literal:

• the person is so tired that they can barely physically keep their eyes open.

But holde øjnene åbne can also be used more figuratively, like stay alert or watch carefully, depending on context.

Examples:

• Hold øjnene åbne! = Keep your eyes open! / Stay alert!
• Jeg kunne ikke holde øjnene åbne. = I could not keep my eyes open.

So here it is a very natural, everyday expression for extreme tiredness.

Yes, but the meaning is slightly different.

• Jeg er meget træt. = I am very tired.
• Jeg er så træt, at ... = I am so tired that ...

Meget træt just states a high degree. Så træt, at ... sets up a result or consequence.

If you want to continue with that I can barely keep my eyes open, then så ... at ... is the best structure.

Yes, there are alternative ways to express a similar idea, but they are not exactly the same.

For example:

• Jeg er for træt til at holde øjnene åbne. = I am too tired to keep my eyes open.

This is very close in meaning, but it presents the idea as too tired to do it, rather than so tired that I can barely do it.

Another possible structure:

• Jeg er træt nok til at falde i søvn med det samme. = I am tired enough to fall asleep immediately.

So:

• så ... at ... = so ... that ...
• for ... til at ... = too ... to ...
• nok til at ... = enough to ...

They are related, but not interchangeable in every context.

Øjnene is tricky for many English speakers because of both the ø sound and the consonant/vowel sequence.

Very roughly, it sounds like:

• ØJ-ne-ne

Points to notice:

• ø is a vowel English does not really have.
• øj is a diphthong-like sequence that can take practice.
• The ending -ene is the definite plural ending here.

The singular/plural forms are:

• et øje = an eye
• øjet = the eye
• øjne = eyes
• øjnene = the eyes

This is an irregular noun, so it is worth memorizing as a set.

Yes, very natural.

A Danish speaker would absolutely say:

• Jeg er så træt, at jeg næsten ikke kan holde øjnene åbne.

It sounds normal in speech and writing, and it is a very common way to describe being extremely tired.

You might also hear slight variations such as:

• Jeg er så træt, at jeg knap kan holde øjnene åbne.
• Jeg er så træt, at jeg næsten ikke kan holde mig vågen.

But your original sentence is completely idiomatic and standard.