I did it -- a bunch of times. I had shaft corrosion eating packing until I replaced the shafts. That boat was previously a liveaboard for many years before I bought it, stainless steel is only stainless in the presence of oxygen (slow water drip), and the log was already too short to cut back to get to smooth shaft.
There was one that was reasonably accessible and I had no problem doing. Water would come rushing out -- and it did really come rushing out. But, once I took a breadth and ignored it and just focused on doing the job I was fine. I loosened the nut, cleaned most of the thread with it leaking a bit, pulled it all the way back, finished cleaning the threads with the rushing water, repacked it, greased the threads, put it back on, tightened it up, and readjusted it after some use. To get the packing out, I used a packing cork-screw tool, some "dental tools" from harbor freight. As for wrenches, I used a wrench sold for the purpose at West Marine (but later seen sold generically at home depot) and large pair of channel locks.
There was another one that I didn't have good access to, no matter what I did. I could barely reach it. I would have to use two plumbing wrenches to get get it apart. I would stuggle getting the threads cleaned up, but with enough patience could. I could eventually get the packing out, but the last row felt like rolling dice over and over again until getting a dozen snake eyes. I just kept poking until I got it. It was a stuggle to get it back together. And, I could never really, truly get it tight enough. I'd have to tighten it up occasionally.
The biggest thing I had to learn was just to take breadth and ignore the rushing water. With all of the water, it felt like should be rushed. But, really, the pumps could handle well more. The pumps did work while I was doing it, but no matter how long I took, the boat never ended up with a full bilge. They'd run while I was doing it, and on and off for a little bit afterward as the last of the water made its way down. No big deal.
My only fear was that I'd somehow break the nut or the threads taking it off and wouldn't be able to control the flow of water until I could get it hauled. To calm my nerves, I kept some toilet wax rings and emergency marine putty on the boat. In the event of a crisis, I felt I could pack it from the outside until I could get it hauled.
I once had a problem with a seacock on a thru-hull with an integral strainer. I needed to replace the seacock in water. Because of the strainer, I couldn't peg the thru-hull from the outside, which I'd done many times previously in other cases. So, I made a hamburger with the toilet wax, put it onto a home depot plastic bag, and pushed it onto the strainer. It worked perfectly, held for a few hours (things got complicated on the inside), and peeled right off cleanly when done. That was in Los Angeles with a water temperature of about 67 degrees. I'm not sure as to how it would have worked in warmer water -- the wax could have been too soft (in which case the denser putty from West Marine may have worked better). Regardless, I felt good that it could be controlled from the outside with toilet wax and by snorkeling from the outside in an emergency.
At the recommendation of an old timer on the dock, I also got an old bicycle inner tube, cut it to make a tube, and then dissected the tube lengthwise to make two long stretchy pieces of rubber. He told me that I could wrap this around a few times to stop the water while I worked on the stuffing box. I tried this this first time, and it worked, but was so bulky it got in my way. I ended up taking it off to work. None-the-less, I kept the two pieces of rubber. In a crisis, it would have been part of a belt-and suspenders approach to pack it with wax from the outside and to wrap it with the inner tube pieces on the inside.
...obviously neither wax on the outside nor the rubber on the inside would work while the boat was running. The wax would fall off and melt. And, the inner tube would make for a total disaster of some kind (rip, shred, melt, who knows). If something happened with the threads or nut, I'd have had to have to take the boat on one engine to the haul out or get it towed.
At any rate, I hate to advise anyone to open an uncontrolled hole in their hull. So, I don't want to tell anyone to replace packing in-water. That seems like really bad advise to give.
So, instead, I'll just say I, personally, did it a bunch of times, how hard it was depended 100% on access, that I didn't have any problems with the nut, box, or threads, it could have ended badly but didn't, I had an emergency plan that seemed decent, and in the end it worked out well for me.