No! That's one of the best things about silver clay. You do not need a studio or lots of space. I do all my work in a small one-bed apartment. The tools don't need much space, and you won't need much space either to work with it. All you will need is:
Any desk (just a small one will do), I work from my dining table
A safe place to fire your clay, just somewhere where you can clear any flammables out of the way if you are firing with a torch. If you have a garden or a porch, great! I do not, so I do my firing on my kitchen counter, specifically on my stove. I can clear anything flammable around and place a firebrick/soldering block on my stove to fire. But please be sensible and do your own assessment of your place - you will be working with open fire!
If you got a mini kiln, even easier. You don't need to worry about open flames, just put it on a heat-resistant surface. If you happen to have a gas stove - great, you can fire fine silver clay on it. More about it in this blog post.
The absolute basics to start working with silver clay are: a non-stick working surface (like a plastic document folder), a craft knife, some sanding paper, a couple of silicon sculpting tools, a small brush, a gas torch (if torch firing), a firing surface, and some polishing papers. You can read more about the tools, both basics and a complete tool guide here.

And that's enough to start creating some pretty silver clay pieces! I was intimidated to start at first as well, but when I did, it turned out so much easier than I imagined!