One method to set up a small apartment closet for manufacturing is to use drywall.
In stick built buildings, you can soundproof the space by adding extra drywall.
First thing you do is add a layer of styrofoam insulation to the walls. Glue it with liquid nails.
In order to avoid attracting attention bring it in in half sheets over a period of time.
While you are doing that you can run extra wiring inside PVC conduit to have handy outlets for your stuff instead of a rats nest of extension cords. You could even just use good quality fused grounded power strips with long cords for this purpose and mount the power strip to the wall.
Next step is to attach drywall or cement board to the wall. Since you need a fire resistant sheet to keep from creating a fire trap. If you use sufficient glue on the styrofoam board, between the existing wall, you can then glue the new gypsum or cement board to the styrofoam to create a foam sandwich. Hold it in place with a couple long screws, make sure you do not drill screws into the wiring.
Glue the styrofoam first. Let the glue cure over a period of time. Gorilla glue works as well.
The screws are there only to hold the board in place until the glue sets. Use thinner gypsum cause it is easier to handle and you will not add too much extra weight to the walls. Note this method will not work with ceilings.
Tape and paint the walls. I have successfully used the instant pre-mix spakle to cover the seams.
Paint with acrylic water based paint. The smell from oil paint will linger too long.
During this time, you can run a flexible dryer duct for ventilation if you can route this without much hassle. Power that with a couple 2 amp computer fans stacked in series.
You will have to sound proof the door as well. Carpet stapled to the outside will work. For the floor, you can remove the carpet, stowing it away for reinstall when you leave. Add in a layer of plywood. If you have nosy neighbors downstairs, you can use carpet underlay between the existing floor and the plywood. I would use at least 3/4 inch plywood. You can use 2 layers of thinner plywood if weight carrying it up stairs is an issue.
Last thing is to install a work bench, and better lighting. LED lights work nice or get one of those really large screw in CFl lights that they use in outdoor lights with the standard build socket.
With careful planning, your shop should not draw much more power than a microwave oven. A mini mill with a Dremel tool head will draw at most 300 watts, and a 3D printer will draw at most 450 watts. A typical microwave uses between 500-1000 watts.
Mounting shelves on the wall above the bench will free up valuable floor,space.
Wile you will not be able to weld in this space, you will be ale to run a mini mill, desktop drill press, mini milling machine, cut off wheel, and other related bench top tools in privacy.