With many devices that do not detect the drive, you may be looking at a failed drive.
With respect to a shop, it depends on the shop and the enclosure. Did you buy the drive/enclosure from them? I would suspect that they wouldn't replace it unless you bought it from them.
If there are files that you want to try to get, you will probably have to open the enclosure. Some enclosures do not come apart very easily, such as the case with the WD My Book drives. If you can get it open, you can possibly connect it to your PC and try that way. Eliminate the enclosure as the source of failure.
Did the enclosure beep prior to the failure? I would guess, not. I have not heard a hard drive beep ever but that isn't to say that they can't. New technologies.....ya know? It most likely is a board within the enclosure that is causing the beep, if it is in fact a beep. In that aspect, you have to isolate the drive away from the enclosure and connect it to a power source. You can connect power to a drive while your PC is on. All it will do is spin up. If it does that, then that's a big step forward. Shut down, connect the drive to a SATA connector and see if Windows detects it. If it does, enclosure failure.
TL;DR You have to pull the drive from the enclosure to troubleshoot past this point. If that is not an option, check your warranty.