Maybe a year ago, I installed four new Energizer Ultimate Lithium AAA batteries into this calculator (bought new in 2009). The batteries, themselves, show an expiration date of 12-2034 (December, 2034). Although I haven't used it for more than several minutes, total, since then, the 50g is now completely drained.
Does this model have a known problem with draining batteries while turned off?
I would suggest clearing the calculator, making sure that there is no program on it or nothing that is questionable, make sure that the battery compartment is nice and clean from any leaks that might have happen and put fresh, recent batteries in it to test again....
If the problem continues, then I would suggest that there is an internal issue. maybe some condensation in the calc body caused some corrosion (this calc is 11 years old after all) or something like that.
In addition to Cyrille's complete reply above, I'd recommend monitoring the health of a NEW set of batteries using the excellent HP 50g "BTMON" (Battery Monitor) program by Cheolho Kim. It displays the batteries' voltage both digitally and as a graphic gauge. It's freely available here: https://www.hpcalc.org/details/6587
One large benefit of BTMON is that it can let you know right away if the "new" batteries that you just installed are in fact strong. Sometimes "new" batteries are not really new, and sometimes batteries are new, but faulty.
Just in case it matters, the HP 50g does have a real-time clock that runs all the time (even when the calculator is turned off), and a trickle of power is also needed to maintain its RAM, so I'd recommend changing the batteries at least once a year anyhow. If you don't use it for a year, then backup its RAM to Port 2 (which is non-volatile flash memory) and remove the batteries to avoid damage from battery leakage, which has been known to happen with every type of battery, even Energizer Ultimate Lithiums.