On days home sick in the late 90's me and my mother would sit and watch episodes of A Baby Story. While it was fun to watch all the different ways women had their babies (as in not just in a sterile hospital room with yelling and beeping machines), I never thought I'd be trying to look for them now. TLC's site is useless from what I can tell as it was in the early days before they were all reality shows. The later seasons got more sensationalized with every family having a huge medical scare which I'm sure got the show eventually cancelled.
While a handful of episodes are on Youtube/Dailymotion and there's random DVD's floating around (like $50 a pop on most) on Amazon, it seems like plenty of these are just gone unless they were recorded off of TV. There was a good 300 episodes as well so if that makes it a lost cause compared to the many daytime reality shows they aired back in the day then I understand. Just wanted to throw it out there since I try to keep track of the few episodes that pop up.