Actually, one of customers (Philips) once used j65nko's suggestion, using PUR someone glued all the cabled to the machine, both internal and external.
It was, by the way, not even a security critical machine, I think they did it to prevent vandalism since the machine was in a (semi)public place ...
In any case, I could have cut the cables and solder new connections to them, so if I would really want to I could have taken out the drive and accessed the data anyway.
I guess you can also glue or weld the case shut so it wouldn't be so easy to open, but then I would still have a circle saw
It does take more time to access the machine, and also more resources and skills, but at this point you have to wonder just how secure is "secure enough".
Personally, I think putting the machine in a room and locking the door would be more secure and a hell of a lot easier than the above "suggestions"