Learn Hardcore Icelandic: Hún býr í húsi.

Because after the preposition í (when it indicates location), Icelandic uses the dative case. Hús is a neuter noun whose dative singular form is húsi.

Why isn’t there an article like “a” or “the” before húsi?

Icelandic doesn’t use separate words for “a” or “the.” Instead it marks definiteness on the noun itself.
• húsi = “in a house” (indefinite)
• húsinu = “in the house” (definite; note the ‑inu ending)

What does the verb býr mean here?

býr is the 3rd-person singular present of búa, which means “to live” or “to reside.” Don’t confuse it with bjóða (“to offer”).

Can you omit the subject Hún and just say Býr í húsi?

Yes, Icelandic verbs are inflected for person and number, so the pronoun can be dropped if context is clear. However, leaving out hún can sound abrupt or poetic.

How would you say “She lived in a house” (past tense)?