Refurbishing the space underneath stairs
This is just standard lath &amp; plaster, which was the classy, professional way to finish walls until after WWII. Prior to that, gypsum board (sheetrock) was considered to be substandard and "cheap." I wouldn't necessarily bother taking this all down, although you certainly can gut it down to the studs.

If you can identify where the studs are, and use screws, not nails, you could fasten sheetrock straight down on top of this and you'd be fine.

On the other hand, there's nothing wrong with a lath and plaster wall. In fact it's thicker and stronger than a sheetrock-finished wall. So if you could just repair the plaster, that might be another good way to go.