Hi, Brad. An environmentally safe, non-toxic stripper should do the job for you. One example is Citristrip and you can find more info at www.citristrip.com, including their "Ask the Experts" section.
While this kind of stripper is non-toxic, it is not without its hazards. It can give you a chemical burn and you must keep the kids out of the work area. You will want to wear rubber gloves and eye protection while working with it.
There are 2 other hazards to be aware of. With that much paint, there is a probability that some dates to before 1972. I don't know about the laws in Canada, but this was the cut off date for lead based paint in the US and I assume that it was about the same in Canada. Therefore, you do not want to sand, or even walk on it really. It's the dust from lead based paint that is one way to have problems with the kids. The other is by injestion (chewing on the paint). As long as the paint is kept in a wet suspension, like in the stripper, it is relatively safe to handle. But scraping, sanding or, again, even walking on it can release the hazardous dust. You will want to check local regulations.
The other problem with strippers is that you are basically trading off the speed with which it works against safety in handling it. There are strippers that will work faster but I am not comfortable working with them. However, it will take probably 4-6 hours for something like Citristrip to work down through 6 or 7 layers of paint. You will probably want to strip every other stair tread so you can still use the stairs while you are working and again, you will want to keep the rug rats away.
The two most common problems are not giving the stripper time enough to work and brushing back over it to give a nice smooth coat. You want to apply it with a cheap, wood handled, natural bristle brush and just glob it on. Brushing it out to a nice smooth, even layer releases the chemical that does the work so the less you do this, the better.
You want to read and follow label directions on any product that you use, but I have also given you a reason to take the kids out to a movie or to the zoo or something. Can't beat that deal.