I don't know if you searched the forums or not but:
http://www.gog.com/en/forum/fallout_series/screen_going_black_read_this_possible_solution/page1 Then Googling around I found something mentioning the possibility of it being due to another program taking the focus from it and the solution to that was eliminating explorer.exe in the background using something like Palette Stealer Suspender
Could also be your antivirus's real-time scanners, for that you would just mark the game directory as an exception or even just turn them off for a bit
Couple other solutions I found was changing the color bit depth to 16-bit or changing your resolution in control panel and actually leaving that window open
Oh and running it in a virtual console although I don't think you should have to
Hopefully at least one of those works, good luck