Are there compliance problems with selling products with exposed PCBs?
This area is a bit of a thicket; in practice you have to hire a specialist test and certification company to do the tests, and they'll guide you with the precise list of requirements.

In the EU, to be saleable a product must have a "CE" mark. The list of requirements is quite long, and mostly deals with avoiding RF emissions. But there is a section on surviving electrostatic discharge, and that's the part that's going to be hard to meet with an exposed PCB. You might be able to do it with PCB conformal coating.

(You also have to make sure there are no high voltages exposed! That and the mains interference requirement is why external power adaptors are so widespread.)

Edit: CE certification is mandatory for electronic products and you will have trouble selling without it; however if you sell your clock as a kit or component you can skip all this. "Explaining" to stores is likely to hit a bureaucratic wall as soon as someone notices the lack of a CE mark.

It is, however, a self-certification ... putting the CE mark on certifies that you believe it to be compliant.

There's no rule banning bare PCBs per se, it's just the ESD rule that may or may not cause problems. You may be able to design an ESD-safe bare PCB.