So, a basic characteristic is, that gadgets have some primary power source and some means to convert/condition it to voltage needed for their working. To be able to run when the primary source is not available, there is some means to store energy, and the "converter" has to cope with the requirements of the storage, too, convert  voltage back from storage to "application", and switch appropriately between primary/storage source and app/app+storage as power consumer.

Primary power source may be:

• ELECTRIC GRID - from our point of view this is an unlimited source, but the need for wiring and long-term uncertainties mentioned previously make it impractical for many cases. Some gadgets need to have a secondary source for case of outage. VERDICT: ONLY CONDITIONALLY.
• CHEMICAL - omitting exotics like fuel cells and bulky/long-term-wise problematic motor-generators (which are still the probably only viable solution for backing up the whole household, but that's for a different discussion), this leaves us with primary cells/batteries. They have a very high energy content per volume, but are prone to self-discharge and even if some of them (primary lithium) may last a decade or two, it's unlikely they will last a century (for CR2032 manufacturers give self-discharge as 1-2% per year, but at the same time many give useful/shell life as 10 years, all this at room temperature). It's the chemical bonds which contain high energy, but that also means that they sometimes break down without the useful output. There are also other materials in the cells (separator, encapsulation), with which the active materials react chemically and interact physically (e.g. clog up passages), and this all is unavoidable in chemical systems (in button cells the active_material_volume-to-surface ratio is very high, that's why they degrade less than high power batteries where things are arranged to thin films and thin layers and then wound up). Simply put, any chemistry is a big NO long-term-wise. VERDICT: NO.
• MECHANICAL - wind and vibration/sound are popular primary sources in the realm of energy harvesting, but the first is outdoor so out of scope here, and the second delivers too little energy in normal household environment (who would want to live in a house which roars and shakes violently?) While there are more such sources (e.g. water flow from rains) they share one unfortunate characteristics, and that is unpredictability. There is one notable exception though - if the source is unpredictable in time, but its occurrence coincides with the energy requirement. For example, the utility (gas, water) monitors can be powered from a small turbine - when the medium flows thus has to be monitored, power is available. There are already existing piezo-generator-based remote switches - they are actuated by hand, and the command is given by depression, which at the same time provides power for the transmitter. And the electric door lock can have a small generator built into the handle, so in case of power outage it still can be unlocked from a mobile phone or using a passive NFC tag, except that the handle has to be wiggled vigorously a few times.. VERDICT: YES, BUT ONLY WHERE APPROPRIATE.
• THERMAL - there is unusual to have enough temperature difference in a household to have a useful thermal generator providing enough power for enough time. There may be coincidencts, like measuring temperature of a furnace; but that does not sound like something very useful. VERDICT: MOSTLY NO
• NUCLEAR - there exists thermoelectric convertors and even direct beta/alpha convertors or other exotic constructions. These all fall very far from common household, even if they may hold the promise for high output and high life. VERDICT: WE WILL RECONSIDER MAYBE AFTER  A CENTURY
• PHOTOVOLTAIC - scalable, available, safe, in room temperature will have high life expectation. Not cheap though. If - as will be in most cases - primarily solar-irradiated, output is available only roughly half a day. Unusable in some places (e.g. cellar/attic/room with no windows, although for some applications it might be enough to power from electric light). VERDICT: YES, BUT USUALLY ONLY HALF A DAY

Storage may be:

• ELECTRIC - CAPACITORS. This is our main focus. They come in a wide variety of constructions; omitting exotics not applicable here they can be split to three cathegories: electrolythics, ceramic, foil. Elyts are attractive for high power density and low cost, but they leak like crazy and involve chemistry (e.g. they degrade more when not used, which indicates that there are things going on inside one does not really want in the long term)  - this makes them unusable for this project. VERDICT: YES, CERAMIC and FOIL.
• MAGNETIC - supraconductive storage. Out of scope. VERDICT: NO
• CHEMICAL - accumulators and batteries of them. The major drawbacks associated with chemical energy storage has been already discussed above, and they still apply here. Accumulators have high energy density but life limited to years  and thousands of cycles (for 100 years we need at least one-and-a-half order of magnitude more, guaranteed). This is the most usual, but for this particular project unusable option. VERDICT: NO.
• HYBRID CHEMICAL-ELECTRIC - ultracapacitors fall in this cathegory. For some surprisingly, they are *not* capacitors, their high energy density comes from chemical bonds. This is reflected in limited cycle time and for some models surprisingly slow charging and low discharging currents. Again, attractive, but chemistry is a long-term-no. VERDICT: NO.
• MECHANICAL - this sounds funny, but this *may* be a viable option. Think of the good old pendulum clock. They have a long and successful development and deployment history (although only in the purely mechanical side of things), and they apparently can be made to "leak" very little. The electromechanical conversion is certainly lossy and low powered devices won't be easy to construct, but it's worth to pursue this path. Flywheels don't sound like an option; prings sound to be too unstable long-term, given the tensions which are in its essence of functioning; but gas pressure and gravitational (weight-on-a-cord) may be viable, if the mechanics is carefully crafted. VERDICT: YES, MAY NEED RESEARCH