I have read:
Wann kann ich mir die Wohnung ansehen?
Why don’t we say:
Wann kann ich die Wohnung ansehen?
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I recognize two problems here: One, how the word ansehen is used within context, and two, the greater grammatical concept of the indirect object. So we'll cover the specific problem first, and then briefly look at the dative case and indirect objects in general.
So, the verb in this case is sich (dativ) etwas (akkusativ) ansehen. This is similar to a verb you have also likely seen: jemandem (dativ) etwas (akkusativ) geben.
sich etwas ansehen translates into English to "to have a look at something". There's another form of this verb, etwas (akkusativ) ansehen which more or less translates to "to view something". When talking, the former feels more natural.
In a sentence, you may have a subject, direct object (the thing that the subject is acting upon) and the indirect object (the thing that is being affected by the action). The indirect object in German will be in the dative case. I suggest you look up indirect objects.