Each time I am in Necropolis, and I talk with Set, I end up getting a fatal error, and the game crashes, causing me to have to end the program. I did finally manage to finish a dialog with him once, but when I came back and spoke to him again, it crashed just like it had been doing all the previous times on my first attempts to speak with him. Anyone have any idea what could be causing this, or a possible fix? Thanks in advance!

Do you get any error messages during/after the crash? Or is it the generic "Fallout has stopped working blah blah" message??
Are you using any mods or is this completely vanilla??

Okay, this time just walking near Set began to initiate dialog with him on its own, and then I got the follow error as it crashed yet again.
dwwin.exe - Application Error
The exception Breakpoint
A breakpoint has been reached.
(0x80000003) occured in the application at location 0x7c901230.
Judas - thanks for the help! Yes, just the standard patches which update the version and then the hi-res patch as well.
(as suggested here: http://www.gog.com/en/forum/Fallout_Series/fallout_1_faq )
*** Update****
Okay, this time, it said:
"Application Error: E\Fallout.exe
The instruction at 004d2580 referenced memory at ccccccd4
The memory could not be written"
Then it prompted me to terminate the application again, giving me the same breakpoint error as above after doing so

Hmmm....try running Fallout in administrator mode. And turn off Windows Defender (if on Vista) and any other AV programs and try again.
If all else fails you might want to try reinstalling the game (back up your saves first) and see if that clears it up.
If the admin option doesn't work you may have a corrupted file somewhere because when I played the original Fallout I never crashed when talking to Set.
DWwin.exe is a Dr.watson process; it seems you have Dr.Watson running in the background. After looking up the name of the file I found that it *may* be a registry error. Try downloading CCleaner a free non malware registry and system tool and run a registry scan with it. Make sure to back up the registry and create a restore point in case anything goes wrong. Then try running Fallout. I hope this helps. Let me know if I can help with any of this if it is seems jumbled or anything.

I know this may be a pain in the fundament, so to speak, but try running the game pure vanilla. No high res patch no FALCHE or anything and see if that works for you. I would hate to have you go through all that and still have no fix for your problem. And make sure to set Fallout.exe to run in Administrator mode in the compatibility menu. Let me know how it all works out for you.

I finally got past the Set dialogue part by mostly avoiding the situation, but then went back to another town, only to find that it still froze when transitioning areas of the town. It has to be a corrupt file somewhere in the install, because I even moved the game to a different drive with the same results, and also tried the suggestions you made, except for reverting the patches altogether...
At this point, I guess I'm going for a fresh reinstall, and hoping my save game data I backed up will not contain any of the problems I had already. If so, guess it's back to the start again altogether...
Disappointing and a little frustrating, but on the good side, at least I know what I'm doing from the start a little better if I have to restart from the beginning.
Thanks again for all of the help, Judas!

Hey no problem. If the game gives you some other problems at least you have some suggestions on what to try in order to solve them :-)

Very true!
At least it's a fun game, and the thought of starting over still sounds like an opportunity more than it sounds like a total loss.

Hell yes. Especially when you have had one too many of those "Damn I wish I did that quest/whatever a different way". Besides, it's Fallout not some run of the mill RPG :-)

Thanks Grah, but unfortunately, none of the things Judas and I discussed actually resolved the problem in a way that allowed for continuation of the game saves I already had when the crashes began. I did a fresh reinstall and the same problem still occurred with those game saves. I had to start over with a new character again, from the beginning, and with a fresh install. If anyone in the GOG staff can offer some additional insight, I would appreciate it. Thanks.

Thank you, Grah - I will try and do that when I have the time to get it all together and send it...