Yes. Yes they can.
Good companies will have measures to ensure customer privacy, all the way up to ridiculous level stuff like keeping servers inside electrically touch-sensing cages with biometrically locked entrances that can only be entered with a customer representative present.
So generally there shouldn't be a cause for concern with any respectable provider.
Then again, running a server at home isn't that bad. My dad did it, he still does it, and now I do, too. We are each others' off-site backup.
The main issue is usually whether you have access to a suitable internet connection. If you want to access your stuff out-of-home, that is.
The hardware can be almost anything. Depending on what you want to run, you usually don't have to be picky. My machine was built, and gets upgraded, using dirt-cheap parts off the used market, always a couple generations behind the latest hardware.
The only thing I buy new are the hard-drives.