Learn Croatian: Imaju velike dugove.

Dug is a short, one-syllable masculine noun. In Croatian, most one-syllable masculine nouns add the infix -ov- (or -ev- after certain consonants) before the plural endings. The nominative plural is dugovi, and here we see the accusative plural dugove.

The adjective here is velike and the noun is dugove. What case are they in, and why do they both end in -e?

They are in the accusative case because they are the direct object of the verb imaju. For masculine plural adjectives and nouns, the accusative ending is -e. Learners often accidentally use the nominative plural ending -i (which would be veliki dugovi) instead of the accusative velike dugove.

Doesn't dug also mean "long"? How can I tell it means "debts" here?

Yes, dug is also the masculine adjective for "long". You can tell it is the noun meaning "debt" here because it is being modified by an adjective (velike) and it uses the noun-specific plural expansion -ov-. If the sentence meant "long debts", it would be imaju duge dugove.

Why is the pronoun for "they" left out of this sentence?