In Eclipse, can I view the files I currently have open in a vertical stack instead of a horizontal one?

tabs can not be stacked vertically. However, CTRL+E (⌘+E on mac) might give you some relief.

Eclipse 3.8

CTRL+G brings up an Editors dialog as shown in this screen capture:

Eclipse 4.3

CTRL+E brings up the pop-up shown in this screen capture:

With ctrl+shift+e you can open an extra window that show you the open tabs vertically.