I am currently without an oven, so am wondering whether some of the recipes that call for braising in a heavy pot in an oven could also be carried out on the stovetop?
Yes you can do it on the stove top. Cook the meat covered on a very gentle simmer (med-low). Depending on the size of the roast, it will probably need about 3 hours. Check periodically to insure that the simmer doesn't get too vigorous
I agree with Monita above and would add that a heat diffuser will help reduce the risk of burning anything on the bottom. There are lots of styles but I like the cheap old tin ones like the one in the picture.
At least in my opinion the cook top is the best place to braise (unless it's an electric range). You can holder a steady simmer and it's easier to check the progress of the pot roast, and perfume your kitchen while you are doing it.
I prefer braising on the stovetop. so yes, you surely can braise successfully without an oven.
Another vote for stove top. I learned to make pot roast (the 'mother ship' of braises for me) that way, and find it easier to control than in the oven. Sometimes I do pop it in the oven to rewarm the next day, or to keep it warm for a short period before serving.