Okay, I figured it out by myself. At the bottom of the Permalinks page (I didn’t scroll down far enough to see this, previously), it said the following:
If your .htaccess file were writable, we could do this automatically, but it isn’t so these are the mod_rewrite rules you should have in your .htaccess file. Click in the field and press CTRL + a to select all.
So what I had to do was to download .htaccess onto my local hard drive using my FTP program (FileZilla), paste those lines of code into it, then upload it back again. This didn’t work to begin with because I didn’t have permission to overwrite the version of the .htaccess file that was already on the server (presumably thanks to iThemes, though fiddling with the settings for that plugin didn’t seem to help), so I had to right-click that in FileZilla and enable editing for the file owner (me). (Basically, FileZilla executed a chmod command on the file for me.) After that, I was able to upload the file.
My website is no longer broken. But… phew. The bottom of a long page is a really bad place to put an error message! Why not put it at the top, where people can see it without scrolling down?