I’ve done it in the reception of an office, we just used a bottle of soapy water and put towels down to catch the excess. If making zero mess is important put a plastic bag on the floor under it and tape it to the wall to catch the run off.
Depends what you’re drilling. Lightweight breeze blocks it’ll fly through in clouds of dust so you’ll need a 2nd person there with a spray bottle to keep it damp. Ours was cement with pebbles/ballast so it was slow going and I could just splash some on myself when it started to dry out.
Why not do what builders do? gaffer tape a polythene curtain(B+Q paint poly sheet or similar) mask up inside.
In general it’s better to avoid the source of the hazards than to just wear PPE to attempt to mitigate it. And if you did make a little shower cubicle to contain the dust it’s going to be everywhere and end up in your eyes too so you’ll need a full mask. Silicosis does not look like a fun way to die so if you can avoid generating dust in the first place it’s much better all round.