@masbass,
Thanks for the info so far. Here are my comments (all IMHO) based on what you've said. Other people may have different views.

masbass wrote:
This drive is used as a backup drive
Without wanting to be pedantic, since you say that you need files which only exist on the drive, that isn't what most people would call a backup (i.e. it isn't
another copy of data which exists elsewhere also). The point is that this news changes the whole risk/reward ratio, if the data on this drive is important to you. Any DIY recovery attempts may cause further / faster deterioration of what is likely to be an already-failing drive. If you are lucky, you might recover most or all of what you need; if you're unlucky or make mistakes, then you won't and your attempts may make any later professional recover harder and/or more expensive or, in the worst case, impossible.
masbass wrote:
just storing files that aren't accessed frequently anymore, and never leaves the house. It's almost always turned off in its USB enclosure and I use it only when I want to copy files there, not as a live, always connected to PC drive. No falls or shocks up to now.
OK, so no hints there what may have triggered the problem.
masbass wrote:
When I go home I'll look at the SMART data and post them here.
Thanks - remember, the full, raw SMART values are needed, in addition to the usual "value, worst, threshold" SMART values for each attribute. As
einstein9 mentioned, some unreadable sectors are very likely - I'm interested to see that info (SMART attribute #5) and other attribute values, to see how severe the drive thinks the current situation is.
masbass wrote:
File copying starts normally, and then after a short while data transfer speed drops and drops and drops and if you leave it copying it even disconnects and you get a device not found from Windows.
OK, that still fits with a possible drive fault.
masbass wrote:
From what I read around it seems many WD green drives like mine had a bug in their firmware and were practically useless (imagine trying to run an OS on top of that mess) but I haven't found any solution to that no matter how much I googled.
Your problem description is so severe that I don't believe you have a f/w fault. However if that is the only path you want to persue, just say so and I'll leave you to it and wish you good luck.
masbass wrote:
I need the data from this disk and will try to empty it before executing any changes (that will take a while...)
As I mentioned above, DIY recovery attempts have risks, which you need to consider and accept before you continue.
You have 2 basic options to try to get the data off the drive - either try to copy the files (if you only need a very small number of files, you
might be lucky doing this although you've explained that your drive won't successfully copy for long), or cloning the whole drive to another empty, working, 1.5TB or larger drive, using software which can cope with (and has some error recovery control regarding) unreadable sectors. There are a few choices for cloning software depending on your budget (some are commercial, some are free), your expectations regarding features (GUI or CLI, supported or not, degree of control etc.), your skills with other OS (some are Linux/Unix, some are Windows, some are DOS etc.). In my (non-DR) work, if I need to do this, I use GNU ddrescue on *nix, but that is unlikely to be suitable for you unless you have some *nix experience, and even then, it is not a "one click" utility. Other previous recommendations from other members include DMDE, Media Tools Pro, HDClone etc. Do not try to use Ghost, Acronis, XXLClone etc. as they are not designed specifically to cope with failing disks. You must get the direction of the clone (source vs. target) correct, otherwise you irreversibly erase your data!
Unless your USB enclosure (which I guess was one you bought and fitted the drive into) does encryption, then in general it is usually more successful to clone when the failing drive is removed from the USB enclosure & attached via (S)ATA instead of USB. The (empty, working) 1.5TB or larger target drive of the clone can be USB-attached, since we don't expect it'll need error recovery to occur during the clone. If your enclosure does do encryption then this needs to be investigated further to decide on your options.
Overall, unless you need only a few files, then cloning the whole drive (or at least the partitions containing data, is that is much less than the whole drive) with suitable software is generally recommended - but you're still taking a risk whatever you do. (I believe that DMDE has some features in this area to minimise what needs to be cloned, assuming that the MFT is readable, but that isn't software I've investigated myself, as I can't use it at work.)
If the data has more than minimal value to you, you may want to consider using the services of a DR company, if they have dedicated cloning devices (many companies do), as those devices will typically do a better (more successful & less risky) job of cloning a failing disk. Your data, your risk, your choice!