Can I remove this closet?
The walls themselves are probably not load-bearing but there may be a post that supports a beam or a beam junction.
The best way to proceed is to hire an engineer before making such a substantial alteration. My bet is, there is sooner a post than a whole wall that bears the load. And, if I wanted to do further homework before hiring a professional, I would look at the orientation of the joists. The walls that are parallel to your joists are unlikely to be load-bearing. Ones that are perpendicular, may be. Both of those wall types may be hiding a post inside. To determine joist direction, remove one of the two ceiling air returns or vents you have there and you will likely reveal the joists. This will save you making unnecessary holes in the wall at this point.
One important heads-up:
I just did a renovation in my house where I opened up a kitchen and living room and I am talking from my experience. I ended up leaving the post in a small wall that I left alone and the plan still looks very open, however, leaving the post was the least of my problems.
The walls I removed were full of electric wiring, HVAC ducts, and even a natural gas pipe. Luckily there was no water plumbing as it would have complicated things even more. It is likely that the builders used these walls as an easy way to route all these things between the levels of your house.
As you confirm the &quot;demolitability&quot; of that wall, be prepared to plan for possibly routing electric, HVAC, and other things that can complicate things immensely depending on how much you are willing to demolish in the process. In my case, I had to run a few new cables all the way from the basement, as the existing wire wouldn't reach to where I wanted it and I would no longer have a room for Jbox either. I had to reroute HVAC ducts under the floor and above the ceiling (in the attic space). If you have access to look under your floor and above the ceiling, you will gain a better understanding of what you are dealing with.
I don't mean to discourage you, it maybe completely DIY-doable (I did and I had no prior experience in the project as involved as this), and you may have a very easy case. Just be prepared for these things as you plan. Ask a lot of questions here before you assume and after your own research. If you think through and plan well, you will likely come up with some creative ways to go around obstacles and still stay to the code. Folks here are amazing and I couldn't have done all this that easily if I hadn't had their help (thanks folks!)
Here is a before-after of the things I had to deal with in just ONE wall.

Here is another where non-load-bearing wall concealed a load-bearing post, as well as ducts.