You can also get the time free from GPS. The big advantage to GPS time (or internet time) is that if you loose the power or the time-signal, it can be automatically re-set... When the battery in your watch dies you have to re-set the watch after replacing the battery. When the battery in your phone dies, you don't have to re-set it after re-charging.
Regular electric clocks (the kind with a motor) use the power line frequency. In modern countries it's super-accurate and the correct it over time so your clock should be "perfect" as long as you never loose power.
Hum (electromagnetic radiation) is going to vary depending on what's powered-on around you so it's going to be unreliable but the frequency will be stable (except for other electromagnetic noise). When you say "portable" the AC radiation becomes more unstable/unreliable.
I would like to sense the AC wirelessly and as part of a portable something. I have no concrete plan yet but I think it would be nice to have "free" very (long time) precise time source.
Although the energy consumed from the power line will be immeasurable, you will be consuming energy and the source is the power line. You'll have an (inefficient) transformer or capacitor.
I did a quick-and-dirty (and possibly inaccurate) estimate of 120VAC across a 1M resistor and I got a little more than 100 Watt-hours per year. That should cost less than 2 cents. You could use a 10M or 100M resistor to reduce that further. And, you'd still need to power the clock.