Learn Croatian: Ovaj posao nudi sigurnost.

You are right that sigurnost is the direct object and is in the accusative case. However, sigurnost belongs to a special group of feminine nouns that end in a consonant (the i-declension). For these nouns, the accusative singular form is exactly the same as the nominative singular form. So, it doesn't get a new ending here.

How can I tell that a word ending in a consonant like sigurnost is feminine and not masculine?

The secret is in the suffix -ost. In Croatian, almost all nouns ending in -ost are feminine. They are usually abstract nouns describing a state or quality, like sigurnost (security), radost (joy), or mladost (youth). Whenever you see that -ost ending, you know you are dealing with a feminine consonant-ending noun.

Posao (job) is the subject here, but I remember it looking different in phrases like puno posla (a lot of work). Why does it change so much?

Posao is a tricky masculine noun because it features two common spelling rules: a fleeting 'a' that disappears when endings are added, and a final o that turns back into an l. Here it is the subject, so it stays in the nominative form posao. But in the genitive case (like after puno), the a drops out and the o reverts to l, making it posla.

Is nudi (offers) an imperfective or perfective verb here?