There's unfortunately no way to restore a library from the storage folder alone (and yes, I'm sorry to say that I am absolutely certain of that).
Did you sync your Zotero library? http://www.zotero.org/support/sync
that would allow you to just sync it back to your formatted PC.
Otherwise, and if you really don't have any other back-up of your old harddisk I don't see an alternative to re-creating your library.

Hi all,

thank you very much for such a quick reaction!

Yes, I did sync Zotero but not sure it was synced at the time when I downloaded this huge quantity of articles that I want to restore (I think my free Zotero account could not even contain them all).

No, I have no other copy of my hard drive, I copied the "storage" folder and pasted it onto the removable USB drive, then formatted the PC...

I am afraid I will have to restore my library by hand. But thanks anyway. Please let me know if you have any ideas about how to make the restore the fastest possible - re-download the bibliographies and then join the PDFs by hand..? (I no longer have access to the database where I downloaded them from). Thanks!

I'd first try to sync and see how much of the data syncs down to your computer--even if you were over your free quota, the data would still sync. Then manually attach (drag) the PDFs to the right item.

You do not need to drag-drop the PDFs to re-attach. Even if your storage quota was exceeded online, the metadata (including information that a file is attached and what that file was) would have synced. You would just need to copy over your storage folder to your new data directory. (I would copy the contents of the storage folder and choose to not overwrite existing files)

If you still have items that you're missing, let us know, there might be some tricks to bring your library up to date with your storage directory.

The drag and drop works, however Aurimas, if you know about a way to avoid it, it would be awesome. What I did was indeed that I pasted my backed up "storage" folder from the old Zotero (without the SQlite file) into the new "storage" folder now on my PC. The problém is that Zotero just does not see the files that I pasted this way. It only sees new files that I downloaded, and those that were synced before.
The copy-paste trick did work last time (at least I think so), but it did not this time... Any ideas?

I don't have time right now to go into details, but if you're missing items, my suggestion would be to do the following.

1. Use mronkko's Perl script from https://forums.zotero.org/discussion/9091/orphaned-attachment-files/#Item_8 but change the line at the bottom from rmtree(["$zoterostoragefiles/$_"]);tomove("$zoterostoragefiles/$_", "<some_temporary_location>/$_"); and add use File::Copy; next to the other use statements at the top. Also <some_temporary_location> should be a path to a folder you create. Running this script would then (in theory) copy all orphaned PDF attachments to this temporary directory.

2. Drag-drop all of the PDFs from the temporary directory (in Windows for example, use a search in File Explorer to look for .pdf) into Zotero (maybe a new collection to make things easier). Select all PDFs in Zotero (Ctrl + A), right-click and run Retrieve Metadata. Fill in metadata for items that didn't get retrieved automatically by hand (or re-add from websites, which may be easier).