Ek is besig om my hande in die badkamer te was.

Besig om ... te is a very common Afrikaans way to express an action that is in progress, similar to English am/is/are ...-ing.

So:

Ek is besig om ... te was
means
I am washing ...

Literally, besig means busy, so the structure is something like I am busy to wash ..., but in natural English we translate it as I am washing ... or I’m busy washing ....

This is part of the fixed Afrikaans pattern:

besig om + infinitive phrase + te + verb

So in your sentence:

• Ek is besig om = I am busy / I am in the middle of
• my hande in die badkamer
• te was = to wash

You should think of besig om ... te as one grammatical frame rather than trying to translate om and te separately every time.

Because in Afrikaans, the main verb often goes to the end when it appears as an infinitive in a structure like besig om ... te.

So instead of saying something like:

• Ek is besig om te was my hande

Afrikaans prefers:

• Ek is besig om my hande ... te was

This is very normal word order in Afrikaans. The object and other information often come before the infinitive verb at the end.

Yes, here was is the Afrikaans verb meaning wash.

So:

• te was = to wash

It just happens to look like the English word was, but it is unrelated in meaning here. This is one of those cases where a familiar-looking word can be misleading.

Yes, my can mean either my or me, depending on how it is used.

In this sentence, my comes before the noun hande, so it is clearly the possessive adjective:

• my hande = my hands

Compare:

• Hy sien my = He sees me
• my hande = my hands

So the position in the sentence tells you which meaning it has.

Hande is the Afrikaans plural of hand.

• hand = hand
• hande = hands

Many Afrikaans plurals are formed with -e, though not all nouns follow exactly the same pattern.

Because Afrikaans often puts extra information such as place, time, or manner before the final infinitive verb in this kind of sentence.

So the structure is:

• Ek is besig om
• my hande
• in die badkamer
• te was

That is normal Afrikaans word order. The location phrase in die badkamer comes before the final verb was.

Yes. That sentence is also correct.

• Ek was my hande in die badkamer can mean I wash my hands in the bathroom or, depending on context, I am washing my hands in the bathroom.
• Ek is besig om my hande in die badkamer te was makes the ongoing action more explicit: I am in the middle of washing my hands in the bathroom.

So the besig om ... te version is more clearly progressive.

Yes, it is grammatical and natural.

That said, in everyday speech, people often choose the simpler version unless they specifically want to stress that the action is happening right now:

• Ek was my hande in die badkamer.
• Ek is besig om my hande in die badkamer te was.

Both work, but the second one emphasizes the ongoing action more strongly.

Die badkamer means the bathroom.

In Afrikaans, die is the definite article for both singular and plural nouns, so it can mean the in all kinds of situations.

Here:

• in die badkamer = in the bathroom

So die is simply the normal word for the.

Not always literally, no.

On its own, besig often does mean busy:

• Ek is besig. = I am busy.

But in the pattern besig om ... te, it often works more like an aspect marker showing that something is currently happening:

• Ek is besig om te lees. = I am reading.
• Literally: I am busy to read

So it is connected to the idea of busy, but in many sentences the most natural English translation is simply the progressive form.