I have two Saints and the previous answers here covered a lot of what you need to know, but maybe I can go into more detail.
Slobber...when they shake their head, they get some good distance and I routinely find slobber on the ceiling, on kitchen cabinets, doors, walls, etc. If you don't plan on crating them or keeping them in a particular room, you can count on surprise areas of slobber on your work clothes and coats from when you return home and they are so very happy to see you..... There is no such thing as a "non-drooling" St. Bernard. You may get lucky and find one with small jowels that slobbers only a little. In addition, it stains rugs and will collect underneath them as they lay on your floor.
One of my favorite sounds is the tails of my two babies banging against the wall when I come in, they are so happy to see me.
Health problems....bloating....you have to be very careful about their activity levels one hour before and one after eating and or drinking a lot. Bloat can take a Saint's life in a matter of hours, it's very serious. It's important that their food and water sources are raised up to mouth level for them. Hip Displaysia....we paid $3,500 for hip surgery on our male....and that was only one side. Fortunately that did the trick and we did not need to have the other side operated on.
Shedding....they shed a lot. I am fortunate to have two short hair St. Bernards and that helps, but as much as I brush mine, there is dog hair everywhere. I stopped buying fleece because the hair works its way under the fibers and is hard to get out.
They do not like to be alone, separation anxiety caused us to get a second Saint.
Having said all that, Saints are great dogs. They are intelligent, beautiful, curious, funny, extremely loyal, and loving. They are a lot of work, but worth every bit of it. I have enough room in my house where we could spread out, but my furkids always choose to be near me. You can just start talking to them or approach them and their tails start wagging with happiness. My male lets out this great snort as he's falling asleep, it makes me laugh all the time. My female loves attention and will nudge both of my hands for love...I can't just use one hand, she nudges them until they are both petting her. It's really cute. They are the best.