Power ESP01 in forest
I think that in this case you will need multiple solutions for generating the power.  


You need to know how much power you need.  This depends on you solution, I can't help there.
You need a backup battery, ideally one that doesn't mind constant trickle charge and deep discharge.  Sealed lead acid are probably the best.  
Your battery must have enough power over a long enough time.
You need a power generator, wind, solar, water, chemical. And they should all have a common interface so you GeoCache is modular, so the GeoCaches can be plugged into any of the power sources, maybe even more than one source.  


The problem with the power source is it is going to be a visual clue to where the cache is, probably also an audio clue.
The battery, if SLA, is going to be a great big lump that could be dangerous if damaged.
The caches are not going to be cheap, just because of the power source and battery, so I hope you have deep pockets.  

Rather than looking for a ESP01+BT4 module look for any microprocessor with BT built in, it maybe a better way to go.