Ostensibly, there are rechargeable cells in the 16340 size of CR123A. There are RCR123A Li-ion cells, which are usually LiCo, and often protection circuit is recommended for a number of reasons. A more recent chemical addition to this size is the IMR16340 cells, which are more resilliant to overdischarge, but really depends on how long the cell was overdischarged, and how it was treated in order to recharge. IMR cells don't come with an option for a protection circuit. I have had a lot of success safely recharging overdischarged IMR cells, and keeping them in service, but just recently, I killed one dead by recharging it without allowing a long enough rest period after use, now won't hold charge.
In reality, and practically, there are no rechargeables that can replace a CR123A in capacity or voltage. The best option I have seen so far is to jump up to IMR 18350 cells, which offer about half the capacity of a CR123A, but still with the increased 4.2V terminating or max voltage. This voltage can destroy some lights that require a 3V nominal voltage, but in regulated lights, it should be an acceptable replacement.
Another option, possibly, is using 2xAA NiMH cells, which will give roughly the same voltage and capacity as a single CR123A... but you take a hit on the size of your battery tube, as 2xAA cells are a lot longer than a single CR123A.