I remember you writing that there were other roomates living with you. Why don't you tell your family that the room is rented. If they get curious about where the tenant is after a week, you can either make up a story about them being away, or just tell them that you don't get into their business as long as they pay on time and don't make a mess. Either way, how could they ever expect you to show another person's personal space without them present? You could say that you're storing some things for a close friend who found themselves in a bad spot from the economy, and once again, don't feel comfortable showing someone else their things. They lost their house, and can't fit everything in the new apt.
I have 4 roomates, who regularly come to my door, and my tent is a huge black spot in my room, very visible thru even a crack in the door. I planned ahead, and set up the tent before I even ordered the seeds. I left the door to the tent open and hung clothes from the frame, even put some sterilite totes inside with clothes convincingly tossed on them to look like they were filled with clothes(they were filled with coco and pumps, etc.). I let everyone see the thing for 2 weeks and ask questions. Told them it was a portable closet cuz I needed more space. Now when they see it with the door closed, they don't suspect anything. Gotta be smart and plan ahead, Imagine what questions you would REASONABLY ask in their position, and find a way to deflect their attention. I'm sure you can pull it off with your attention to detail. However, like MB, I feel that if your family is,for some reason, so unreasonable as to demand entry to a private area of your house, then you have no choice but to be just as unreasonable. If they can act that way, who says what they will do if you concede and show them? The result could be much worse than just the end of your grow!