Can I install data wiring in a clothes closet?
The current solution being installed in new homes is a "structured media enclosure" that you can install between the studs in a closet wall. Since it's between the studs, you need to make sure nothing is currently running in that space today (although an outlet that's wired from below servicing the other side of the wall would be very convenient). You want this on an interior wall so that you're not dealing with insulation. The outlet is installed as part of the enclosure and the enclosure is covered to reduce the risk of clothing blocking vents and starting a fire.

Here's a sample enclosure:



See this question for photos and a discussion about typical wiring in one of these enclosures. Your patch panel for networking, phone splitter, cable splitter, router/modem are all installed inside the panel. Typically, electrical is wired from below and all the data wiring goes through access holes in the top. When closed, no wires come out the front of the panel, everything stays inside the wall and panel.

I'd also recommend installing an access panel in the ceiling of your closet to make it easier to fish new wiring, unless there is access from above. If you're going to open up a lot of walls/ceilings to fish the wiring, then you may also want to install some conduit to each of your drops in case you want to change this network in the future without opening everything up again.