Just to indicate that not only Swedes think it perfectly reasonable to talk about buying a service, I am an American (admittedly elderly and academic, but not an economist) who has seen, heard, understood, and probably even used myself just such a construction, without even giving it a second thought.
I agree. As a native English speaker, I think context would be necessary for me to understand either "I'm buying a service" or "I'm buying a favor", but both seem equally grammatically correct. Here's an example of the expression being used in a national publication: "The rationale for including the domestic clause was to prevent a state—or even Congress—from buying a favor from the president."
In American English we say we "buy" something if we can take possession and ownership of an item, otherwise we "pay" for it. This is made somewhat murky by recent "terms of service" agreements that only assign license to items, like music or software downloads. This is why for digital items they'll use the verb "purchase" instead.
I don't understand this. There was this other lesson I was going through just before this one (I can't remember which one), and there I translated "the service" as "tjänst" and it gave me an error. The correct answer in that specific context would have been "Servicen". I looked up in discussion and it was mentioned that Servicen is the service such as one you get in a store and officialy (from post, from a clerk, etc...), while tjänst is more of a favor, something you ask of a person to do for you. This made me understand that service is a service and tjänst is a favor. So now I come here and suddenly tjänst is not a favor anymore (as it was again shown as a false answer) so you're not buying a favor, but a service - the service that by definition is "Servicen" and not "tjänst" - which I was told to be wrong about in the last lesson. So it means you can't buy "tjänst", you can only buy "service". So, is there any chance these contexts can be adopted and synchronized between lessons, so I don't get told the context of these two words means one thing in one lesson and completely other thing in another lesson? Thanks.
Edit: We now accept buying a favour as well, see Stephanie's answer above. Original answer below.
English "service" has several meanings, and they differ somewhat in Swedish. For the "favour" sense, you can use tjänst, but we don't accept it here because you can, by definition, not really sell favours. :)
Going the other way, the Swedish tjänst can also mean different things. For the kind of service you buy - such as a taxi drive or a haircut - it's a perfect fit with the English service. If it's the service of e.g. customer service, then it does not work.