Learn Croatian: Želim sebi priuštiti taj skupi kaput. - I want to afford that expensive coat for myself.

sebi is the dative form of the reflexive pronoun sebe. It is used whenever the pronoun refers back to the subject of the sentence. Since the subject is 'I' (želim) and the action is being done for 'myself', we use the reflexive sebi.

Could I use the short dative form si instead of sebi in this sentence?

Yes, you can use the short (enclitic) form si, which is very common in everyday speech. However, enclitics must go in the second position of the sentence. If you use it, the sentence becomes: Želim si priuštiti taj skupi kaput. Using the full form sebi adds a little more emphasis to 'for myself'.

Why does taj skupi kaput look like it is in the nominative case?

It is actually in the accusative case because it is the direct object of the verb priuštiti (to afford). However, kaput is a masculine inanimate noun (a thing, not a living being). For masculine inanimate nouns, the accusative case endings are exactly the same as the nominative case endings.

Why is the adjective skupi used instead of the short form skup?

skupi is the definite form of the adjective. You must use the definite form here because it comes right after the demonstrative word taj (that). Whenever you point out a specific object using words like taj, ovaj (this), or onaj (that over there), the adjective must take its definite form.