Yes, visiting Saudi Arabia is probably more difficult than anywhere else in the world, but it's not impossible by any means.
First up, if you're a citizen of a Gulf Cooperation Council country, the answer is easy: you don't need a visa to travel to Saudi Arabia.
If you're not, you have four (well, three) choices.
- Actual tourism visas have been suspended for time being, even for group travel. They'll probably change their mind again at some point again, so might as well keep an eye out.
- Business visas are available if you can find a company in the Kingdom to sponsor you for one and pay the rather stiff application fees. Once you're in, you're in and the Kingdom is your oyster, except for the Muslim-only zones of Mecca and Madinah. Contrary to popular belief, business visitors do not need an exit visa, that only applies for long-term work visas. And if your wastah (connections) are strong enough, anything is possible: I've met single women and Jews in the Kingdom on these visas.
- Transit visas are available if you have a legitimate reason to pass through the Kingdom on your way to somewhere else. Booking a Saudia flight with a layover of over 18 hours is one option, driving a car through Saudi from eg. Bahrain or the UAE to Jordan is another. Plenty of stories on the net about this, here's one documented in extreme detail.
- Last and least, there's one foolproof way to visit the Kingdom: book a Saudia flight with a layover of under 18 hours, in which case you don't need a visa. Only problem is, you'll also be stuck airside at either Jeddah, Riyadh or Dammam airport, which I can tell you from personal experience are all excruciatingly boring. (Survival tip: Mooch free wifi from the unsecured network at the al-Fursan lounge.)
If you're Muslim and male, you have two extra choices:
- A Hajj visa, available for visiting Mecca and Medinah only during the month of Hajj. Visas are allocated by a lottery system based on the population of Muslims in each country and are generally quite hard to get. Wikivoyage has some more details.
- An Umrah visa for visiting Mecca and Medinah outside the pilgrimage month. Saudi Arabia has recently announced an "Umrah-plus" visa that would also allow travel outside the holy sites, details on this are still scarce at the moment.
Neither option is available to women, who must be accompanied by a mahram (male guardian) or, if over 45, travel in a group.
And a personal note: I spent around half a year in Saudi Arabia on business, and from a visitor's point of view, it's not as bad as reading some of the clueless drivel on the net makes it sound. If you use common sense, respect local custom and don't do anything blatantly stupid, you'll be fine. Now, for some of its inhabitants Saudi is considerably worse than you'd think (for example, I was astonished by the number of destitute beggars on Riyadh's streets), but that's another story.