Learn Hardcore Turkish: Çamaşır makinesi bozulsa bile elde yıkamayı öğrenmemiz gerekiyor. - Even if the washing machine breaks, we need to learn to wash by hand.

bozulsa bile – slightly more hypothetical, more like even if it happens to break down (lighter, more tentative).
bozulursa bile – more “regular conditional”, like even if it breaks down (when/whenever that happens).

In everyday speech, bozulsa bile is very common and sounds natural here. The overall meaning (even if it breaks down) stays the same.

What exactly does bile mean in this sentence?

bile is a focus particle that usually means even.

In combination with the conditional:

bozulsa bile = even if it breaks down

Outside of conditionals, bile still means even, placed after the word it emphasizes:

biz bile = even us
o bile yapar = even he/she does it

Here it emphasizes that even in the case where the machine breaks down, the rest of the statement still holds.

Why is it elde? What does elde literally mean?

elde is el (hand) + -de (locative suffix: in/on/at).

Literally: elde = in the hand.
In this context, it is an idiomatic adverbial meaning by hand (as opposed to using a machine).

So:

elde yıkamak = to wash by hand
elde bulaşık yıkıyorum = I am washing the dishes by hand

It does not mean “in the hand” in a physical sense here; it functions as a set phrase.

Why is it yıkamayı and not just yıkama?

yıkama is the verbal noun / infinitive form: washing or to wash.
But in this sentence, it is the object of öğrenmek (to learn), so it takes the accusative case:

Root: yıka- (to wash)
Infinitive / verbal noun: yıka-ma → yıkama
Buffer letter: -y-
Accusative suffix: -ı → yıkamayı

So:

yıkamayı öğrenmek = to learn washing (by hand) / to learn to wash (by hand)

Turkish often marks the verbal noun with accusative when it is the specific thing being learned, wanted, etc.

How does yıkamayı öğrenmemiz gerekiyor work grammatically?

Breakdown:

yıkamayı – the washing / to wash (object in accusative)
öğren-me-miz – our learning
- öğren- = learn
- -me = verbal noun (learning)
- -miz = our (we)
gerekiyor – is necessary

Literally: our learning to wash (by hand) is necessary.

Functionally, this equals:

We need to learn to wash by hand.

So X-memiz gerekiyor / X-mem lazım is a common pattern meaning we need to X or it is necessary for us to X.

Why is the person marker -miz on öğrenmemiz and not on gerekiyor?

In this construction, öğrenmemiz behaves like a noun phrase (our learning). The verb gerekiyor stays in 3rd person singular, because:

Conceptually the subject is “our learning to wash by hand”.
Turkish expresses the person on the verbal noun (öğrenmemiz) instead of conjugating gerekiyor.

So structurally it is:

[Elde yıkamayı öğrenmemiz] gerekiyor.
= Our learning to wash by hand is necessary.

This is a very typical Turkish pattern for necessity and obligation.

What is the role of gerekiyor here, and why not gerekir?

Both come from gerek- (to be necessary).

gerekiyor = present continuous form (literally “is being necessary”) but usually just means is necessary / is needed in modern Turkish.
gerekir = aorist; can sound more general, factual, or formal, often used for rules, principles, or timeless truths.

In spoken Turkish, gerekiyor is very common for practical, current necessities:

Gitmem gerekiyor. = I need to go.
Gitmem gerekir. = I should go / it is necessary that I go (a bit more formal/general).

In your sentence, gerekiyor sounds like a concrete, practical need, which fits well.

Could we also say öğrenmeliyiz instead of öğrenmemiz gerekiyor?

Yes, but the nuance changes.

elde yıkamayı öğrenmeliyiz
- -meli / -malı is the “should/must” suffix.
- This directly says: We should/must learn to wash by hand.
elde yıkamayı öğrenmemiz gerekiyor
- Literally: Our learning to wash by hand is necessary.
- Slightly more neutral / impersonal, like stating a requirement.

Both are natural. -meli/-malı often sounds a bit more direct as advice or obligation; öğrenmemiz gerekiyor sounds a little more like it is required / it is necessary that….

Why is gerekiyor in the present continuous (-yor) if the meaning is more general?

In modern Turkish, the -yor form is often used not only for literal ongoing actions, but also for:

current states
current necessities or obligations

With gerekmek, gerekiyor is the default way to say is necessary / is needed right now or in the current situation, even if it is a general rule for the near future.

So although it looks like a continuous tense, in everyday use it simply expresses present necessity.

What does çamaşır makinesi literally mean, and what is the -si ending?

çamaşır = laundry, washing
makine = machine
makinesi = machine + -si, a 3rd person possessive form

In compound nouns, Turkish often uses this possessive suffix on the second word:

çamaşır makinesi – literally the machine of laundry → washing machine
kahve makinesi – coffee machine
çamaşır suyu – bleach (literally water of laundry)

So -si here does not mean “his/her machine”; together çamaşır makinesi is a fixed compound meaning washing machine.

Can I change the word order, for example: Elde yıkamayı öğrenmemiz gerekiyor, çamaşır makinesi bozulsa bile?

Yes, Turkish word order is relatively flexible, and that alternative is possible:

Çamaşır makinesi bozulsa bile elde yıkamayı öğrenmemiz gerekiyor.
→ Emphasis slightly on the condition.
Elde yıkamayı öğrenmemiz gerekiyor, çamaşır makinesi bozulsa bile.
→ Emphasis more on the necessity; the condition is added after.

Both are understandable and correct. The original order is a bit more neutral and common.