Learn Russian: Я не могу позволить себе этот дорогой компьютер.

Yes, exactly! In Russian, there isn't a single direct verb for "afford". Instead, you use the phrase позволить себе, which literally translates to "to allow to oneself".

Why is it себе and not себя in this sentence?

Себе is the Dative case form of the reflexive pronoun себя (oneself). The verb позволить (to allow) requires the Dative case for the person who is being allowed something, answering the question "allow to whom?".

Could I use мне instead of себе since the subject is Я?

No. In Russian, whenever the person receiving the action is the exact same as the subject doing the action (I allow myself), you must use the reflexive pronoun себя (or its case forms like себе). You cannot use мне here.

Why do этот дорогой компьютер look like they are in the Nominative dictionary form if they receive the action?

They are actually in the Accusative case, because they are the direct object of what you are allowing yourself. However, for masculine inanimate nouns like компьютер (and their matching adjectives), the Accusative form is identical to the Nominative form. This is why they don't change their endings.