How do you archive a Company?

There's not a way to archive a company the same way you can archive a deal, but you can achieve something similar using custom fields. 

For example, you could create a custom dropdown field that has two options, "active" and "archived," and make all the companies currently in your account "active." You could then set up a filter on the Companies list view page to only show "active" companies. When you want a company to be hidden from that view, you could change it to be "archived."