Battery pack solution for a 5V 500mA project
So, do 8000mAh Li-Po elements exist? If they do, it seems they are extremely rare.
  


They're not rare. They're just expensive. Batteries that size are used fairly frequently by RC model airplanes and if a cell of a certain size does not exist then they put them in parallel. As mentioned, larger mAh come with increased safety issues. The chargers to charge such large batteries in a reasonable amount of time are also quite expensive, and such chargers also tend to need a separate high current AC-DC power supply which further increases cost.

Batteries with that capacity are used to fly not-exactly-small airplanes and helicopters which would suggest your went wrong somewhere in requiring such battery capacity for a much less intensive application. You don't even need this comparison to know something is wrong. Your system is idling 92% of the time but that is consuming 80% of all your energy. Your "resting mode" current is a whopping 220mA. That is far too high.

What is consuming all that current when your system is supposed to just be sitting there doing nothing? Deal with that and your battery will be 1/5th the size. Don't let things idle with power for no reason. Put things into sleep mode after a period of inactivity or power things down with MOS switches until there is a reason to power them up.

Stepping down is also easier than stepping up, so I would suggest you go for a two cells in series to get 7.4V and step that down to 5V.