The closest thing to having your own place in the setting of the Matrix universe is having your own base. In the missions we see that the organizations have numerous resources in the city. Redpill Extraction Points, bases, safehouses, listening posts, labs, etc. Though I agree with Zeroone that having an apartment isn't too nonsensical, especially in the post-Revolutions era.
But as people have said, there's currently no feature in the game that allows you to officially take over a building or room.
Saleskawa9 wrote:still be cool though :smileyhappy: