I found the solution!
Sorry for the necroposting, but I think this might be useful for those who will google a similar problem and find this thread, as I did.
zorba2357, I faced the same problem. I tried many things, including file manipulation, deleting one of my family members, but the problem remained.
It was like the whole sim family was cursed! Even if I moved to a different house, the game crashed every time I saved it (even though it was saved). The problem is that it becomes impossible to go to the city, because when loading locations, the game saves the family and crashes.
But in a few hours I found a VERY strange solution to the problem.
The problem is in the file UserData\Neighborhood.iff.
This file contains information about all the families and their relationships.
So, you need to open this file using a third-party utility IFF PENCIL 2 (easily found on Google), and then:
1. Click on FAMs in the list on the left
2. Find your family in the list (you'll have to open each one by double-clicking, you'll find it by name).
3. Change the family name! Anything but what it was. You can just delete one letter to try it out.
At this point, I launched the game and everything worked!
I don't know what happened to my name, but it's like it's cursed and if I change it back, the game crashes again.
But in general, the simpler solution is, of course, to take regular backups.