Learn Croatian: Stipendija zahtijeva barem trideset bodova u jednom semestru. - The grant requires at least thirty credits in one semester.

Particles like barem generally sit directly in front of the specific word or phrase they modify to show their scope. Here, it restricts the quantity trideset bodova. If you moved it before the verb (barem zahtijeva), the scope would shift to the verb, meaning the grant 'at least requires' the credits (implying it might do other things, but this is the minimum action it takes).

Can I use the shorter word bar instead of barem here?

Yes, bar and barem mean the exact same thing and are completely interchangeable in this context. bar is just a slightly more colloquial, shorter version, but both work perfectly to set a restrictive minimum.

The verb zahtijeva (requires) takes a direct object. Why is the word for credits (bodova) in the genitive case instead of the accusative?

This happens because of the number rule. Any quantity of five or more (like trideset) forces the noun that follows it into the genitive plural. The number trideset acts as the object of the verb, and it overrides the accusative case that the noun would normally take.

Why does the timeframe use the locative case (u jednom semestru) instead of the accusative?

When referring to a fixed, static period of time during which a condition must be met, the preposition u takes the locative case. You would use the accusative with u for days of the week or when expressing physical motion toward a destination.