How safe are pre-charged supercaps compared to pre-charged batteries?
How safe are pre-charged supercaps compared to pre-charged batteries?
  I'm designing a solar-powered system and I'd like to use something like a Maxwell's BCAP0005 supercap (2.7V, 5F) to store a lot of energy.



This supercap is far less energetic and far less dangerous than a battery pack of 2500 mAh AA cells or even than an individual AA cell. 
The supercap that you have chosen would produce a burst of power for a fraction of a second that slightly exceeded what you would usually get from a modern 2500 mAh NimH AA cell - but it would then be fully discharged. 
A single AA cell would however produce somewhat less power but for many minutes. A single cell or a few of them in a battery would be quite capable of creating very high temperatures and starting a fire. The supercap could be used to start a fire only with great difficulty.
Some supercaps or ultracaps are far more capable than this one. Some can be used for eg automobile starting.


" .... pants on fire" - almost: On "a few occasions" I have come close to setting my trouser pocket on fire or of burning myself with the temperatures generated  by (stupidly) carrying a number of AA NimH cells (under 2000 mAh capacity) plus coins and keys in the same pocket and having a conductive path form. Removal of the cells from the pocket suddenly becomes one's sole and overwhelming priority with removal of trousers a close second choice. This is not an experience that I intend to replicate in future :-). On such occasions some coins or keys achieve temperatures well above their safe handling level in a few seconds. The supercap chosen as an example could not achieve this result.



First, let's look at the energy capacity aspects:
By any normal meaning of the term, a 2.7V, 5F capacitor will not store "a lot of energy". If all the energy in the capacitor was available it would provide E = 0.5 x C x V^2 Joule. For a 2.7V, 5F capacitor E = 0.5 x 5 x 2.7^2 =~ 18 Joule.
 By comparison a 2500 mAh AA battery will provide about
 E = V x Ah x 3600 Joule
 = 1.2 x 2.5 x 3600 = 10,800 Joule.
 So the 5F supercap will store about 18/10,800 =~ 0.17% of the energy in the battery. Also, whereas the battery will be able to deliver almost all this energy in a typical application, the supercap will need extra effort to recover energy as the voltage approaches zero. 

Discharge safety:

Where a supercap is useful compared to a battery is in it's ability to charge rapidly and to discharge rapidly, and to do so over many more cycles than a battery can with little loss in capacity.
 in SOME cases this rapid charge &amp; discharge capability is large compared to that of a typical battery, and is extremely large compared to that of a battery when expressed in terms of their total capacity. 

However, the example that you have chosen, and all the other members of its family, are far "wimpier" than some super/ultra caps. The data sheet available here shows internal resistance of 170 milliohms suggesting a short circuit current of around 2.7/0.17 =~ 16A when fully charged, and the typical  short circuit current is shown as 16A at 65C and 14A at 85C. As temperature would rise rapidly and equivalent voltage (once de-shorted)  would fall very rapidly under short circuit (as at 18 J gross capacity the s/c discharge time would be well under 1 second) this capacitor would not produce vast amounts of energy (maybe 20 to 30W peak) and only for well under 1 second.

A typical modern NimH AA cell contains  much more energy and will deliver much much more power for much longer. 

For example a 2.5 Ah Nimh AA cell may be typically discharged at up to about 5A and may have a loaded discharge capacity in the 10A to 20A range for short periods at very reduced voltage. So it's output Wattage almost matches what the supercap will produce for a fraction of a second, BUT the battery will produce 10+ Watts for many seconds and 5+ Watts for many minutes.