Electrically they'll be safe unless you do something really stupid such as having both red and black terminal wires touching the bare metal of the cabinet when the amp is on. Even then you're unlikely to get a shock because the voltages involved are low (less than a couple of 9V batteries) and the amp will short out before you do any serious damage to yourself.
You'll know when the amp shorts out. There's a bang and a flash followed by that puff of magic smoke escaping. At that point its game over for the amp. Then it's "Hello council tip".
If you asked me where's a really bad place to install speakers, then embedded in a metal filing cabinet would be right up there with the worst.
Apart from the simple logistics of finding sufficient space without fowling a drawer runner or drawer itself; and then the hassle, the mess and the deafening din of cutting out in a metal box, not to mention the jagged edges and piles of metal filings that will find a new home in stuck to the speaker magnets, and in the cones which then makes the speakers buzz horribly, and in all your files, on the floor, in your clothes, embedded in your skin etc etc..... you have to do all of that in a room which you say is very crowded.
Once you have them installed (and before many of the metal filings find their way to the speakers to wreck the sound even further) when you play music it's going to be accompanied by the rattling and buzzing of a big metal box.: Not nice.
If you have a bee in your bonnet about doing this then no amount of logic will sway you from your course. However, there are alternative and better solutions.
One area that's unlikely to be crowded is the ceiling. Have you considered installing speakers there? If you're prepared to go through all the mess and hassle installing in filing cabinets then a ceiling install would be a walk in the park by comparison.
