В библиотеке можно взять какую угодно книгу.

Because в библиотеке means in the library as a location.

With в:

• в + accusative usually means motion into somewhere: в библиотеку = to/into the library
• в + prepositional usually means location: в библиотеке = in the library

So here the sentence is talking about what is possible at/in the library, not movement to it.

Russian often uses impersonal constructions where English would say you can..., one can..., or it is possible to...

Here, можно is an impersonal word meaning:

• it is possible
• it is allowed
• one/you can

So В библиотеке можно взять... literally feels like In the library, it is possible to take..., but in natural English it is usually You can take/borrow...

Because можно is followed by an infinitive.

So the pattern is:

• можно + infinitive = can / may / it is possible to do something

Examples:

• можно читать = you can read
• можно войти = you may come in
• можно взять = you can take/borrow

That is why you get можно взять, not a finite verb like можно возьмёшь or anything like that.

This is a very common question, because it is about aspect.

• взять = perfective
• брать = imperfective

In this sentence, взять is natural because it refers to a single complete action: taking/borrowing a book.

So можно взять книгу suggests:

• you are allowed to take a book
• you can check out a book
• the action is seen as one complete event

If you said можно брать, it would more likely suggest repeated, general, or process-like action, for example:

• В этой библиотеке можно брать книги домой.
  = In this library, you can borrow books to take home.

So both are possible in some contexts, but взять fits this sentence very well because it focuses on taking a book as a completed act.

Какую угодно is a free-choice expression. It means something like:

• any
• whichever you like
• whatever kind of ... you want

So какую угодно книгу means:

• any book
• whatever book you want
• whichever book you like

It is stronger and more explicit than just a plain indefinite form. It emphasizes freedom of choice.

Because the phrase is the direct object of взять, so it must be in the accusative case.

The noun is книга:

• nominative: книга
• accusative: книгу

The word какой must agree with it:

• nominative feminine singular: какая
• accusative feminine singular: какую

So:

• какая угодно книга = any book as a subject
• взять какую угодно книгу = to take any book as an object

That is why both words change:

• какая → какую
• книга → книгу

No. In this expression, угодно stays the same.

The variable part is the question word:

• кто угодно = anyone
• что угодно = anything
• где угодно = anywhere
• какой угодно = any kind / whichever

So in какую угодно книгу, only какую changes to match case, gender, and number. Угодно remains fixed.

They are very close, and in many situations both can be translated as any book.

But there is a slight nuance:

• любую книгу = any book, often the simpler and more neutral option
• какую угодно книгу = absolutely any book, whichever you want, with stronger emphasis on free choice

So какую угодно often feels a bit more emphatic, as if saying no restrictions — choose whichever one you like.

In a library context, взять книгу often naturally means borrow/check out a book, even though the basic verb взять literally means to take.

Russian often uses everyday verbs like взять in contexts where English prefers a more specific verb such as borrow.

So in this sentence, depending on the context, natural English could be:

• You can take any book in the library
• You can borrow any book from the library

Both may be possible, but borrow is often the most natural library translation.

Yes, Russian word order is flexible.

В библиотеке можно взять какую угодно книгу is natural because it starts with the setting:

• In the library...

This gives the location first, then the statement about what is possible there.

You could also hear:

• Можно взять какую угодно книгу в библиотеке.

But that version can sound a little less neat, and in some contexts в библиотеке might feel more closely attached to книгу, as if describing where the book is.

So the original order is a very natural way to present:

1. the place
2. the possibility
3. the object

Yes, but be careful: the meaning changes in an important way.

Нельзя взять какую угодно книгу usually means:

• You can't take just any book
• You are not free to choose absolutely any book
• there are restrictions

It does not usually mean You can't take any book at all.

If you want not any book at all, Russian often uses ни:

• В библиотеке нельзя взять ни одной книги. = You can't take a single book from the library.

So:

• можно взять какую угодно книгу = you may take whichever book you want
• нельзя взять какую угодно книгу = you may not take just any book; choice is limited

It is normal, natural everyday Russian.

Nothing in it is unusually formal:

• в библиотеке is ordinary
• можно is extremely common
• взять книгу is everyday phrasing
• какую угодно is standard and natural

So this is the kind of sentence a learner could really hear or use.