That's a question I've asked myself a few times, too.
The present-day standby instruments use small LCD screens too, which still rely on some kind of separate secure electrical supply, and presumably so does the attitude reference (gyro of some type).
As far as i know both Boeing and Airbus use a Thales system as standby instrument. In the 737 the ISFD (integrated standby flight display) has its own internal battery pack that provides power for 150 minutes completely independent of the rest of the aircraft, its own internal inertial sensors which even act as the third inertial platform during fail operational auto land operation. The only inputs it receives is heading information from the left IRS and ILS information from NAV 1.
According to our maintenance those units have a much lower failure rate than their old mechanical counterparts, not surprising as they have no moving parts. Additionally those old units were powered by the general aircraft batteries which only provide enough power for 60 minutes. Inertia keeps them running for a bit after power is off, however the attitude reference becomes unreliable very soon after power is removed, unlike the ISFD which runs happily on its own even after the rest of the instruments fail because the aircrafts battery is depleted. In that case you need a flashlight to read the magnetic compass for heading reference though. Additionally it uses the same presentation as the PFDs, has no parallax error and is therefore easy to use from both seats, and has its own light-source of course which keeps it usable for those 150 minutes in darkness as well.