Xbox says that I've run out of space when I clearly haven't

I do use my console online. ... Is there any way to fix that?

You could try the following:

Take a game that you might not care about the save game.

Usually, since your console is connected online, it should have your save games safely stored in the cloud, and should be able to downlad it again to the local cache if needed. But if that part is corrupted, you might lose the savegame, so try it with a
game you don't care much about first.

Highlight the game under "My Games and Apps" and press the Menu button.

Choose to manage the game.

There on the left side, choose to clear the local savegame from that game.

Repeat that with several other games.

Now try if you can start your game with the freed up savegame cache. Make sure that the game is saved and properly synchronized with the cloud.

I do use my console online. While I don't play Fallout 4 with mods at all, I uninstalled the game just to be sure, and it didn't fix the problem—so I can only assume that means that the system cache is corrupted. Is there any way to fix that?

Only a month ago? I've had mine for as long as I've had my Xbox—about four years—and while I don't really use a lot of its space, I also highly recommend using one, as I haven't had any issues with storage (up until now).

If there's an issue with the system's cache, then hopefully a power cycle of the console can help sort that issue. There's a guide on how to do that at this
link.

Are you part of the Xbox Insiders by any chance? I've had issues with installs before and it's been tied to one of the preview updates having an issue.

Hi! These are the suggestions I've been receiving here. I also briefly consulted this page before writing the question—none of the steps listed have worked. Thank you for the response.