One of the possible methods that Destiny could use is a more advanced form of Subspace Capacitor. In The Daedalus Variations, the Daedalus utilized a subspace capacitor developed by Dr. Rodney McKay to extract energy from subspace in a similar manner to a ZPM, therefore providing comparable levels of power. Two crucial differences make this a likely possibility that this is the type of technology behind Destiny.
First, most of the methods the Ancients used to harness Zero Point Energy for power generation were notoriously dangerous. Project Arcturus (SGA S02E06), and The Trinity Experiment (SGA S03E08) come to mind. These projects utilized Zero Point Energy extraction, which were able to generate enormous amounts of power. However, this extraction method created exotic particles which inherently destabilize space-time. In The Daedalus Variations (SGA S05E04), there is no such mention of Zero Point Energy extraction; instead drawing power from subspace, a region of space-time folded away from our four dimensions on the plank scale.
Second, and the more crucial difference, the Subspace Capacitor had the ability to recharge. Zero Point Energy extraction is assumed to be irreversible, so the Subspace capacitor must use some other mechanism of generating power that is. A fascinating observation is that Destiny's mechanism of power storage decays over long time scales (what doesn't?), resulting in Destiny's power storage systems' diminished functionality.
Rush: Probably about-- less than 40% of original design capacity
It is likely that the Ancients had experimented with a multitude of different methods extracting vacuum energy from subspace. Each probably had different advantages and disadvantages, such as the ability to reverse the the flow of energy (remove entropy from subspace at the expense of increasing it (i.e. heat) in "real" space), or how quickly energy can be extracted. While it is curious why the Ancients in the latter stages of their civilization decided to utilize non-rechargeable forms of portable energy storage (ZPMs), it is not necessarily indicative of a deliberate choice not to utilize subspace capacitor based ones. It may simply be that the technologies involved in Destiny's construction were simply forgotten, or buried underneath layers of archived information in the Ancient Database.