Housing a Power Supply
These power supplies are not safe to leave exposed due to mains wires connections, they are meant to be inside an equipment enclosure, but they do need quite a bit of airflow. So a small enclosure with no air circulation won't cut it.
If you can put it in your lathe stand, that would be practical, since that would make the whole device self-contained, easier to move or store without a power brick attached. But if it fits inside a small compartment it will need a fan, and if this jeweller's lathe is used to cut metal, it will also need an air filter to prevent metal shavings from ending up inside the power supply. Make sure the airflow goes inside the power supply, not around it.
If the motor is brushed and uses a simple PWM controller, it should be possible to control the fan from the same controller.
You could also use a power supply that doesn't need a housing.

Although these have rather short wires, so you still need to put the wire nuts somewhere. You could also use a readymade brick rather than build one:

This will most likely be cheaper than a separate enclosure for your power supply.