Hi.
Well, I sadly don't know what kind of case it is actually.
There are those four rubber things in the corners at the bottom side of the unit so it doesn't slide around on the desk, I found by removing one of those that there are in fact screws hidden underneath.
The problem I am having is tthat I haven't done this before and I am a bit afraight of touching something which shouldn't be touched or do anything wrong by accidentally breaking something.

Greetings Moritz.

Hi.
Oh dear, so as we found out the things I thought to be screws were'nt actually screws, the problem is that the casing of the hard drive is actually glued together as it seams and there is no way to open it, the only way I could access the drive would be flexing it open or ask a sighted person to use a saw, cut a part of the case away and reach the drive that way to get it out of the case and in to a new one.

Greetings Moritz.

Having done this multiple times, I can say with every confidence that it's a pretty easy job. You'll want a nice bick flathead screwdriver. Since the case isn't of use any more, don't worry about damaging it.

You'll want to find at the edges where the seams for the case are. Force your flathead into that seam, then pry the hell away.
Every external I've torn apart, and I've torn apart about five of them, has been an internal hard drive inside a case, with a small controller board to operate it. Tear away the outer plastic shell. You'll often discover that inside is a metal casing around the hard drive. This casing is sometimes held together with screws that go into the holes which would normally secure the hard drive inside a computer case. Remove this case.
You'll often find the hard drive is connected to the controller board by some connectors. If these are wired connectors, simply unplug them and you're set. If these are connectors on the controller board itself, firmly pull the drive away from the board in a nice straight movement and it'll come loose.

Once you've done this, you can install the hard drive like any other internal, or you can buy an external hard drive enclosure to put the salvaged device in.

Remember while disassembling: be gentle with the hard drive. There are moving parts, and you don't want to be banging it around in case you damage some internal workings such as drive heads.

Kai

WARNING! Read this before beginning!

I've recently heard stories of people moving their drives to new cases, only to find their drive works, but the data is unreadable. This is because the old enclosure used encryption on all the drive's data, with the key stored on the control board of the enclosure. Presumably this is to insure that only the original enclosure will work, but it ends up biting people who have enclosures die. If this is the case, the enclosure may have the only copy of the key that exists anywhere in the world, in any form. Alternatively, it might be that the manufacturer can recover the key for a hefty price. In either event, if you find your data is gibberish, OS doesn't recognize the filesystem on the drive and software data recovery programs are of absolutely no use, this has probably happened to you. Without the encryption key, sadly your data is as good as gone, as modern encryption systems are generally very good at what they do, with no known backdoors or weaknesses, and make a brute force search of the key take probably millions of years.

Bottom line: Do not throw away your old enclosure until you know you can actually get to your data on the drive with a new enclosure. The drive powering up is not enough.