Can I build this surfboard in my apartment/living room/dorm room?

That depends on whether you live alone. Your board may take as much as two full work weeks to build IF you were working on it nonstop. Chances are you are more likely to do a little at a time every day - plus weekends - for a couple of months. Be sure that your mom/wife/husband/roommate is cool with having your new stick growing in the middle of the room for that long. Good luck with that. You'd also be wise to disclose to your loved one that you will be generating an incredible amount of wood shavings, saw dust and sanding dust which, though aromatic, is not good to breathe. Let’s not even talk about the epoxy resin which will drip off the sides of the cloth onto whatever floor covering you forgot to spread a drop-cloth over. You will need enough room for a rocker table longer than the length of your board, with at least two feet of space around all sides, plus workbench or table-like surface to assemble parts on.

And good lighting is essential - when you begin shaping and sanding, you will want it at the same level as the board itself so that you can see any imperfections.