The process for building bases has changed in the NEXT update.
Starting a new base
Just find a spot you like the look of and build a Base Computer. It's in the quick build menu, z on PC, I believe is d-pad down on a controller.
Build your case computer, and interact with it. This will kick off the process, where the game will allow you to claim the area. After this your build menu will include all your base items and you're ready to start construction.
Recalling an existing base
You will need to obtain Navigation Data items, either from finding them in Outposts and containers, or buying them from the Galactic Trade Network. Buy quite a few, about 30.
Go to your chosen planet and build a Signal Booster from your quick build menu. Place this and interact with it. Select the option to use 1 Navigation Data, and proceed to scan the planet for 'Habitable outposts'. Hopefully you'll see 'Shelter found' and it will be marked as 'Habitable base'.
You are looking for a Habitable Base to show up. Just keep trying until you find one. If you feel unlucky, pickup your Signal Booster, and fly to another area.
Once you've identified the spot, fly over there, and you should see a huge flat open area, just like in the Foundation update. There should be a Base Computer in the middle of the area.
Interact with this base computer and you'll be able to restore your base.
Video for reference
It seems that existing bases are re-loaded 'as-is' meaning all your storage is restored as well.
The process is detailed in this Reddit post.
Answering my own question here, it turns out to have been, if not a glitch, at least a bad interaction due to a corner case.
What happened was that, after I had placed the Agricultural Station in my base, I was editing my base structure from outside. I briefly misplaced a structure so that it overwrote the Agricultural Station, but since I was outside, I didn't notice.
Apparently the result of this—since I'd already activated the Farmer part of the mission and it had told me to go find it—was that the Farmer showed up in the system, but jumped away from the place—station or planet—I was on so I couldn't reach it.
Once I went back inside my base (having given up, for now, on the Farmer), I noticed the Agricultural Station was missing.¹ I reconstructed it, went back to the station, and:
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¹ This may be useful for some: if your inventory is filled in the process of deleting an element (because you have space in your inventory for the main item, but not subitems like glass for windows or you fill a stack when there's no slot in your exosuit to start another), you can build the Base Salvage Capsule, which is a large machine like a Large Refiner. It will contain any such items. (It can hold a huge amount, but allows only a one-way transfer to your exosuit.)
Best Answer
So it seems this is a glitch that can fix itself. I logged out last night and logged back in earlier today and the 'Grave' marker was on my radar. Flew over to the location and all my inventory was there.