It really depends on what you buy for the camera. This can be done with a simple web cam from Best Buy attached to a PC with software running that either streams the video to the server, or steams a string of .jpgs of the image at some interval.
Or you can buy a more expensive camera that has all that built in, and instead of attaching it to a PC, it plugs into a router port. You then "log in" to the camera via your web browser and supply server details to the camera. Settings within this type of camera allow you to specify what to upload and how often. In some cases you can even send a video stream to one server and a .jpg to another. (hint hint...this route is definitely the way to go)
One of the best sources of information I've found is Ambient Weather's guide "Weather Camera Review, Purchase and Installation Guide" (yes it's true they want to sell their camera's and they only seem to sell higher end IP cams, but there is a reason for that Also the info in the guide is generic enough and covers all levels well enough to be a learning tool for you as well as a camera sales tool for ambient.)
To answer your specific question, I tried at one point, but was never happy with the result (using a low resolution camera looking out a window). I will do again when IP camera prices fall further, and/or I have more discretionary income floating around to "pay the price to do it right".
As far as dreamhost concerns, uploading a constant stream of .jpgs shouldn't get you into any trouble even on shared hosting. Bandwidth is unlimited and there is nothing that should run memory usage up, unless you're station becomes very popular and traffic to your site does that for you. However I can't comment on resource usage for steaming live video.
My very favorite weather station with live steaming cam is this one: http://weather.reefradio.com/ out of Florida. There is a contact link at the bottom, and those folks are really friendly, I've traded an email or 3 with them.