Deep 13

According to lore, Deep 13 is actually located below Gizmonics Institute. There is a Deep 12, which housed some foxy neighbors for Dr. Forrester and TV's Frank. Dr. F's movie watching experiments were housed here, and Deep 13 was also where the broadcast controller for these experiments (called Mystery Science Theater 3000, in order to trick cable companies into thinking it was a program rather than an experiment) was housed. Deep 13 got a facelift in season 6 when Dr. F's mother came to visit, and it kept this new look until its final episode 706: Laserblast when Dr. F's funding was cut (funding?) and he was forced to move back in with his mother, leaving the Satellite of Love and it's crew to their own devices.

Former Best Brains employees have said that the Deep 13 set was so large that they couldn't keep it constantly standing like the SOL bridge, so they assembled it fresh for every shoot, and tore it down afterwards. Also, when the movie came out BBI sold copies of the flick and included a piece of the Deep 13 set with each one as a promo.

• It's apparently close to the atomic pile. Dr. Forrester finds this an attractive feature about the place and likes being down there.
• Despite being a secret underground lair, lots of people found their way down there during the course of MST3K's run, including pizza delivery men, the mob, Jack Perkins, Lisa Loeb, and many many others.
• There's more to Deep 13 than what's seen on TV. The lair includes bedrooms, a shower, and a kitchen. While these areas were never seen on the show, sound effects have revealed their existance (such as the sound of running shower water, the mention of preparing a full Thanksgiving dinner, and TV's Frank talking about cleaning the bedrooms).
• Deep 13 is littered with Dr. Forrester's previous experiments. The buildup increased over the years until being cleaned up during an audit in experiment 512: Mitchell.
• A long tube called the Umbilicus connects Deep 13 to the Satellite of Love. Items can be sent back and forth between Deep 13 and the SOL with relative ease.
• The only Satellite of Love crew member to visit Deep 13 after being trapped on the SOL is Crow, who was shot down the Umbilicus in an attempt to find the secret button to bring the Satellite of Love back to Earth. However, he bumped into Dr. Forrester before he could accomplish anything and jumped back up the Umbilicus to the SOL.
• In real life the Deep 13 sets were very cumbersome. During filming a wall would often fall down, often onto the props or the actors (as shown on the MST3K Poopie tape).
• The infamous button (as in, "Push the button, Frank.") was stolen from Deep 13 during the 1996 convention tour. Seeing as how the Deep 13 sets were going to be torn down anyway, it wasn't replaced.
• Best Brains sold a line of t-shirts in 1996 with the Deep 13 logo and the words "Deep 13 Athletics" printed on them.

Information comes from watching MST3K for years and years. Bragging time: I have a piece of Deep 13 in my display case.