I'm in a small apartment too, where my partner (and dog) are also here. We've both got corners set up in different rooms (I'm in the bedroom) and if I'm in my corner, I'm working. It's less about having a full room to dedicate to being a home office, and more about making a space (no matter how small or open) dedicated to working.

I would suggest trying headphones. Noise canceling ones if you have them. Makes it easier to ignore the distractions around you.