How do you hide or archive a note in Evernote?

No there is no simple built-in function for this.

But there are multiple ways around it.

You could make a notebook 'Archive', where you move notes into.

OR

You could tag "archived" notes with a tag 'Archive' and then whenever you search, use "-tag:Archive".

EDIT: If you want the contents of the note to not be visible in the 'Notes by'/'All Notes' section, then you can encrypt them.

You can export them to mhtml or html, then delete them from your Evernote database. Later you can open them in a browser, word processor, or have Evernote scan the folder to import them. The nice thing about this solution is that you can still search them with Windows Search or any search utility. Or you can zip them so they're not in your way.

The Export options are in the File menu, or by selecting your desired notes using the ctrl or shift keys in your list, then using your right-click menu to export.

MHT/Mhtml is the most complete format and will save the truest backup without messing too much with formatting and scripts. But html opens up in many more programs, and is smaller. Firefox can open mhtml if you get the "Mozilla Archive Format" extension. (It can also associate your htm/html pages with Firefox so you never have to see IE again!) :-)