The two X2 capacitors in the BBC Micro power supply are susceptible to failure due to age and can often go "pop" and let out a plume of smoke. Whilst this display can appear quite scary, it's not a big deal to fix.
In October 2010, my original BBC Micro started to make a rather acrid smell and I managed to turn if off before the capacitors actually went "pop" so I was lucky. Quite often you see BBC Micro's for sale on ebay where the seller declares the that their BBC was fine for half an hour while they were testing and then it went up in smoke. These capacitors are the reason. The problem is far from fatal and the BBC Micro can operate without these capacitors fitted as they are used exclusively as noise suppression. However, without them you may experience interference in other devices so it's worth replacing them rather than just removing them.
Dismantling the PSU is reasonably straightforward and replacing the X2 capacitors is easy if you're handy with a soldering iron but if you're uncomfortable with working on PSU's, there are places online that will service, repair or replace your PSU for you and it may be better that you contact one of those places rather than attempt the "surgery" yourself. If you are still thinking of tackling the swapping out the X2 capacitors yourself then it's also worth considering buying a revised BBC Micro Service Manual from RetroClinic as it has a full section dedicated to servicing the power supply unit.
At this point, it's worth stating that unless you're really confident about changing the capcitors, you should HEED THE WARNING LABEL...
After dismantling my PSU and de-soldering the two X2 capacitors, the picture below shows what I found. The capacitor on the left is the original 1982 X2 100nF capacitor rated to 250VAC. The capacitor on the right is the modern equivalent that adheres to new EU regulations and is therefore rated to 275VAC. The old capacitor has clearly given up