Learn Russian: Студентка сняла жильё на весь семестр. - The student rented housing for the whole semester.

When you state how long an action was intended to last as a result of a verb (like renting, borrowing, or going away), you use на plus the Accusative case. If we just said весь семестр without на, it would mean the physical action of renting itself took the entire semester to complete!

Doesn't the verb снять usually mean 'to take off' (like clothes)? Why is it used here for renting?

Yes, снять (the perfective form of снимать) literally means 'to take off' or 'to remove.' However, it is also the standard verb used in Russian for renting an apartment, a room, or housing (жильё). Note that if you are the landlord renting your place out to someone else, you use the verb сдать instead.

Why isn't the ending of весь changed here, since it follows the preposition на?

It actually is in the Accusative case! However, because семестр is a masculine, inanimate noun, its Accusative form looks exactly the same as its Nominative form. The modifier весь (whole/all) simply matches this inanimate pattern, so it looks unchanged.

Is жильё a countable noun like 'apartment'? Can I put it in the plural?