Swapping the SSDs for blank (or at least empty of anything-you-care-about) ones would work, though of course then you'd have to swap them back afterward and that has some potential to mess up the cabling (depends on your case). It also requires some SSD(s) you don't care about.
Actually encrypting the drive is a good option. You can use Bitlocker (the Windows built-in disk encryption feature), though depending on your hardware and your Windows edition you might need to upgrade some stuff and/or change some settings to make this work (by default, Bitlocker wants a TPM, and I'm not sure if it's available on Home editions or not). You might be able to use "Device Encryption" even on Windows 10 Home; this is just BitLocker under the covers, but with less user control over its behavior (and in particular, it does require TPM 2.0, which many but not all modern machines have built into their CPUs). Alternatively you can use a third-party encryption utility. Veracrypt (among others) is able to perform full-volume encryption (same as Bitlocker) such that the OS and data are inaccessible without a password or USB key. In either case, the shop would still be able to turn the computer on, though not to get very far; they'd be stopped at or before the Windows login screen. However, all the data (not just specific files) on the disks would be encrypted at rest.
Finally, depending on the shop and the machine in question, you could remove the SSDs while whatever else gets fixed, and then bring them with you and ask the technician to install them while you watch. Installing SSDs takes very little time in most cases, especially for a trained and experienced technician, and can be done without ever letting the SSDs or machine out of your sight. Of course, if the SSDs are SATA and the shop doesn't know you're going to install a bunch of SATA SSDs, they might not run the power and data cables where you'll need them, which might slightly increase the time to install them. Not by much, though. Replacing the PSU on most cases really is quite easy, with cable management being the only tricky part at all and still not that hard (unless your case layout is really inconvenient or the cables are too short, which sometimes happens).