Well, first of all, the active verb in your sentence is "to be," which, among others, comes in the forms "was" (past tense), "had been" (past perfect tense) and "has/have been" (present perfect tense). The verb "to buy" is an infinitive.
To answer a specific vocabulary issue you had: The terms "buy" and "purchase" do not describe any ongoing ownership or use of an object. They refer only to a transaction, and a change in ownership of an object. Note that even when the purchase really is a sort of ongoing activity (like when a person has a mortgage), a reference to "buying the house" would still be commonly understood as the time that the mortgage was first arranged and the owners attained physical possession of the property.
Now... Without any other context, the conventional way to form this sentence would be
was able, meaning that it simply happened on a specific and particular occasion, sometime in the past.
The past-perfect tense would only be used if the sentence was part of a larger passage where you were talking about something else that happened in the past, and then you needed to connote that the time that Andrew was able to make the purchase was before even that. So, if you were telling a story about how Andrew invited your parents over to see his new house, you might say
he had been able to buy a house while you were giving the context for the story.
The present-perfect could also be correct under the right circumstance. It describes an ongoing activity or condition. So if Andrew hasn't actually made his purchase, but, sometime in the past, did attain the capacity to do so, then
he has been able to buy a house. If you're presenting Andrew's purchase as part of a list day includes many present-perfect activities as part of one unified indication of his status, you might find it acceptable to match the tense and refer to the purchase in the present-perfect along with actual ongoing conditions. Finally, if the singular act of making a purchase is habitual, it might then also be present-perfect, such as with:
Thanks to my parents' help, Andrew has been able purchase food.