It's weird that Unix is often regarded as something old and dated, often by consultants trying to sell you on Windows. Often they'll try and make out that it's too hard to use, too arcane, and generally a pain, and that you should be using something more modern. But modern Unixes are powerful, flexible and highly mature operating systems that are less bloated than operating systems half their age and can be turned to virtually any task.
Put it this way - can you name any other software that has been around as long as Unix and is still in heavy use today? I can think of one, and that's the C programming language, which was to a large extent developed alongside Unix, and in which most Unixes are now written. But I can't think of any others.
(Note: I'm going to be slightly controversial here and in this context, I'm treating all Unix-like operating systems as Unix. Dennis Ritchie, one of the creators of Unix, has said that he regards Unix-like OS's such as Linux as de-facto Unixes, and I'm going with that here.)
If you consider the desktop market, then it's only really the OS with the largest market share (Windows, which I'm treating as one OS for all the versions) which isn't Unix-like. The number two, Mac OS X, is a certified Unix, and then all the Linux distros are Unix-like, as are the BSD's and OpenSolaris. In fact, you have to go pretty far down the list to get to one that isn't Unix-like - probably ReactOS, FreeDOS, or Syllable.
The various Linux distros are generally doing OK at winning market share away from Windows, as is OS X, partly because people just didn't like Vista. Now Windows 7 is out this may stall for a bit, but the release of ChromeOS later this year should mean a few more Unix devices get sold.
Unix also has a majority of the server market, and always has done - it's well suited to use as a server OS, and has an enviable security record. It's also very useful as an embedded OS - many routers use Linux or one of the BSD's for this purpose.
Finally, Unix has been breaking into the mobile OS market for some time now. Embedded Linux-based OS's were used on a few high-end mobile phones for years, and now Palm have developed webOS, which is also Linux-based. Android is shaping up to be a huge force in mobile OS's, and of course the iPhone uses a cut-down version of Mac OS X.
So, for a crusty old OS, Unix seems to be in remarkably good health. It's well-placed to grab a huge piece of the mobile OS market, and has a respectable slice of the desktop. Anyone tells you Unix is old-fashioned, they couldn't be more wrong. Unix has stood the test of time, probably better than Windows has.