Since it has been a while since you posted your question, I'll go ahead and answer this one.
I'm going to give you a short answer up front, but for the sake of sharing knowledge, I wanted to do more than just copy and paste an answer. The short answer is this:
"keep fish" is "balık yetiştirmek"
However, it appears that some sources (e.g., Tatoeba) indicate that "balik tutmak" conveys the same meaning. See my search for "keep fish" here: Tatoeba: "keep fish. The word "yetiştirmek" appears to have a connotation of "grow" or "raise," so despite what is found at Tatoeba, it may be the better word to use. I think if you use "akvaryum" (aquarium) or "balik havuzu" (fish bowl but literally "fish pool"), you will be understood.
So, how do you say, "They want to keep fish"? I'd probably say it like this:
Onlar balık havuzunun içinde balık yetiştirmek istiyorlar.
Literally, "They want to keep fish in the fish bowl." If I'm wrong, perhaps someone will correct me. [Don't know if I'm "wrong," but this may not be the best way to say this. See LucBE's suggestions below.]
In the meantime, it should be obvious by now that I am not a native speaker of Turkish, nor am I even some degree of fluent, so I rely on a variety of resources to help me out. Many of them I had never heard of before until I began reading the duolingo discussion threads. When I am attempting to translate a word, here's an approach I often take:
When I am trying to translate just a single word, I usually start my search with WordReference.com. I do that because it lists words somewhat similarly to the way a standard dictionary would -- with most common meanings for the word listed first.
If I'm trying to translate something made up of two or three words, however, I usually go to Tureng. It is more comprehensive and seems to be more up-to-date with the most modern words and phrases. It has other extremely helpful features as well. For example, it labels words with categories such as "Computer," "Technical," "Gastronomy," et cetera. It isn't available for every language, but it does include Turkish, German, Spanish, French, and English.
If I need to translate something longer than two or three words, I'll go to Tatoeba. People refer to it as a "sentence dictionary."
For the record, don't think I'm not a fan of Google Translate. I am, but when it comes to Turkish, it is usually a last resort, not a first one.
If you want to take your translations to even higher levels, then consult with the Turkish dictionaries. Nişanyan Sözlük and Türk Dil Kurumu (TDK) are two of the best. You also may want to try Wiktionary. These three resources give you details on the origins of the word which are tremendously helpful in understanding the word better and deciding if it is the word you want to use.
Hope that helped.
My pleasure! :-)
Regarding your suggested translation:
If you use içinde, wouldn't that stress too much the fact that you keep the fish inside the bowl? As opposed to outside of it? I mean, doesn't it sound awkward?
Is Onlar balık havuzda balık yetiştirmek istiyorlar correct?
Or shorter Onlar akvaryumda balık yetiştirmek istiyorlar?
Perhaps I am stressing it too much. It would make sense that the use of a full word rather than a suffixed ending for case would, indeed, seem to add more emphasis. I like your suggestions. I'll leave mine up just so that others can see what you are referring to, but direct readers to your post. Very well done, LucBE!