As barry pointed out, the average power out can't be more than average power in (minus losses, as no process is 100% efficient; often the efficiency is MUCH lower)
However, I think you know it, when in your original question you compare the task to storing water.
Depending how much energy you need to store, you should select storage medium to fit the task: A rechargeable battery can store a lot of energy. A capacitor does not store that much, however they are generally easier to use and have definitely potential for smaller losses. A so-called super-capacitor is a special energy storage capacitor, which may have capacity of several Farads in a small component, so they are somewhat in between battery and "normal" capacitor in amount of energy they can store.
Inductance stores also energy, but it is not good for storing energy for any extended time. Many switchmode power supply converters use that storage, but only for extremely short pulses.
Actually I think the greatest challenge is low input voltage, independent on your energy usage pattern (how often and how long you would take energy out from the storage). In my experience converting from 1.5V to higher voltage without a lot of losses (bad efficiency) is bit challenging. There are switchmode regulator chips suitable for that, but the efficiency is never that great.
You also asked if you can store energy in passive elements. The answer is in principle yes. However, you need also active elements and some of other stuff to convert 1.5V to 3V and to control the system. And the task is quite demanding, starting with your requirements, even in the best case. Especially with such a low voltage and small current as your available energy source combined with significant load requirements.
-ted