I can't make any progress in the game because I am unable to save. The 'Escape' key is supposed to bring up a menu where I can save but pressing it doesn't do anything. I've pressed every single key on my keyboard, in fact, and not a single one of them brings up a menu where I can save the game. This also prevents me from quitting the game in the standard way. I can't Alt+Tab or Windows key my way out the game, either--I actually have to press and hold down the power button on my computer and do a hard restart just to get out of the game.

I'm slowly getting used to the difficult and dated control scheme this game has, but until I resolve these issues I won't be able to make much progress.

AFAIK, this bug happens, when you load the game in the middle of some specific animations (usually - with the "help" of Quick Load via F9).
So just try not to do it again.
Also, you may use Alt+Ctrl+Del and kill Gothic process in Task Manager instead of "hard restart" with Power button.

I just started playing Gothic yesterday and was running into this issue a lot. Here's what I think I figured out about why it was happening and what I had to do to get past it.

Basically, I was trying to load a saved game during an animation or scripted scene, as punknotyet had suggested. In my case, I would get frustrated when I died or lost a lot of health during a fight, so I would try to Load Game-- except it seems like the cursor keys can be sensitive, and I was accidentally selecting New Game instead, so I would try to ESC back to the menu so I could Load Game as soon as the game restarted, and then when the saved game loaded I would be stuck in place-- couldn't move, couldn't bring the menu up again to reload again, etc. I could turn in place, but not move or access the menu again.

I would have to CTRL-ALT-DEL and restart my computer, but then when I launched the game again it would usually give me a black screen or maybe a black screen with an empty window on it. I struggled to fix that issue, and the solution was to open the game's folder in GOG (Manage Installation, Show Folder), go to the SYSTEM folder, right-click GOTHIC.EXE, select Properties, Compatibility tab, check the option to run as an administrator, and apply the change. Next, right-click on GOTHIC.INI, edit in Notepad, change RECALC=0 to RECALC=1, change SCALEVIDEOS=1 to SCALEVIDEOS=0, save, and then launch the game again. That usually took care of the black screen issue.

If you accidentally hit New Game when trying to Load Game, just keep pressing ESC as many times as needed to speed through the opening video and the initial scripted conversation with Diego, then END your conversation with Diego as soon as that option comes up. Once he starts walking away, you can Load Game and everything should be good. So the real solution is to avoid hitting New Game by mistake in the first place, but if you do then just try to fast forward through the new game sequence instead of trying to load your saved game in the middle of the new game sequence.