Do you have another base? Then you can switch the actual model of your base to the new Space theme. You do it from the Decoration menu at the door of your base, using the Relocate option. If you don't have a base you need to finish a mission from your journal. It becomes available at level 21 or so.
You can switch an existing base (if you have one) to the Space theme - using the Relocate Base command on the existing bases' control panel. Note though the space base is a miniature one without all the rooms that many other base themes have - if you relocate the base from an existing one, object placements might not be remembered on switching the base if the existing base is a larger base theme. You can also buy a deed from the marketplace and add it as a second or additional base. If you don't have a base yet, a mission will drop while levelling that gives you one, and you can use the space theme then if you like, though I'd probably recommend one of the standard sized base themes (deco, gothic, or others off the marketplace) as a first base.
]# Just to add, if they have Legendary, or have bought the access to those, then they can. If free-to-play, they can only have a base as standard. Also, it's called moon "base" because it's a base, the bigger ones you're talking about are lairs.
Yes, only membership or premium access with the bases/lairs upgrade bought, allows base use. Forgot about that, it's worth mentioning. On what they're called, Last I knew the game called the standard sized ones bases or lairs (depending if you're a hero or villain), and the smaller ones like the Space theme and the Cave theme 'hideouts'. I forget what the game called this Space theme before consuming it, but I suspect it called it a Space Base in the name just to rhyme.
If you have the base theme applied, when you place the base on top of the building or on the ground you're magically transported to space...somehow. Probably for the same reason that in the 31st century, we still have cell phones and more lamely, telephone booths strewn about.
Level 10-12 (as marked in the journal) for the mission and as said you need to have lair system unlocked. I like to call them hovels....