Learn Russian: Мы сами мыли борта корабля. - We washed the sides of the ship ourselves.

Борт is a masculine noun that belongs to a special group taking a stressed -а in the nominative plural, rather than the standard -ы. This is an irregular pattern you have to memorize, similar to дома (houses) or глаза (eyes).

If борта is the direct object being washed, why isn't it in the accusative case?

It actually is in the accusative case! Because борт is an inanimate noun, its accusative plural is exactly the same as its nominative plural. So the special -а ending is used here for the accusative too.

Why use сами instead of себя for "ourselves"?

Сами (the plural form of сам) emphasizes who performed the action: we did it personally, by ourselves. Себя is a reflexive pronoun used when the action happens to the subject. If you used себя, it would mean you washed your own bodies instead of the ship!

Why is корабль in the genitive case (корабля) here?

The genitive case is used to show possession or belonging, translating to "of the ship" (the sides OF the ship). Also, note the stress shift: the nominative is корабль, but in the genitive, the stress moves to the ending (корабля).