Learn Arabic (Modern Standard): أيام العمل طويلة.

The singular is يوم (day). أيام (days) is a broken plural. Instead of adding a regular suffix to the end, broken plurals change the internal letters and vowels of the original word. It follows a very common pattern, similar to how ولد (boy) becomes أولاد (boys).

Why is the adjective طويلة (long) feminine singular when we are talking about plural days?

This is a strict rule in Arabic: plural nouns that do not refer to humans (like days, cars, or books) are treated as grammatically feminine singular. Since أيام (days) is a non-human plural, its matching adjective طويلة (long) must take the feminine singular form, ending in the taa marbuta (ة).

The English translation says "The work days", so why doesn't the word أيام (days) start with الـ (the)?

أيام العمل is a possessive phrase (called an Idafa), which literally translates to "days of the work." In Arabic, the first word in this type of pairing never takes الـ (the). It automatically counts as definite because it is connected to the second word, العمل (the work), which does have الـ.

Where is the word for "are" in this sentence?

There is no word for "am", "is", or "are" in present-tense Arabic sentences. You simply place the subject phrase (أيام العمل, the work days) right before the describing word (طويلة, long), and the word "are" is naturally understood.