Base building is now a thing in No Man's Sky, thanks to the recent Foundation Update. Typically enough, there are limits to what you can build: you're relegated to a large but still-somewhat-stifling region, and building upwards will only get you so far. That is, unless you make a change to one of the game's files.
As this helpful tutorial explains, you only need to edit the tkgraphicssettings.mxml file, which is hidden in BINARIES / SETTINGS within the main No Man's Sky directory. If you set the property "RemoveBaseBuildingRestrictions to "true" then you'll be on your way, but as always, you do so at your own risk.
I've not done this myself, but there are examples of some of the resulting shenanigans circulating on reddit, in particular this one:
Some redditors are questioning whether it'd be possible to build a connecting tunnel between two planets. Since someone has already travelled between planets using a jetpack mod, I'd hazard a guess that it's probably possible. Possible but, you know, a lot less efficient than just flying between planets.
I've been playing a little bit of the update and am finding my initial No Man's Sky cravings are returning. Chris offered up his impressions yesterday, and you can read them over here.