The answer is two fold. Yes software that isn't dependent on Mac OS 9 hardware drivers is able to run in Mac OS X's Classic environment. If you have software dependent on non-Mac OS X native hardware drivers they won't be able to run on today's new machines:
If you are planning on buying a new machine, it is highly recommend you upgrade your software first to Mac OS X native versions (link describes steps to upgrade, software and hardware which is compatible, and what to do if you aren't ready to upgrade) and then buy a new machine.
In addition it is very important to make the distinction of the two types of Mac OS 9 that exist:
1. Booting ("start up" as called by article 86209 above) into Mac OS 9 - which from Mac OS X is done by selecting Mac OS 9 System Folder from Apple menu -> System Preferences -> Startup Disk and hitting restart. This gives you the full compatibility of Mac OS 9 compatible drivers. Restarting back into Mac OS X is done through Apple menu -> Control Panels -> Startup Disk. This is NOT called Classic.
2. Classic - the application that allows Mac OS 9 to run as a shell on top of Mac oS X. Apple menu -> System Preferences -> Classic lets you control this function within Mac OS X. It only gives you partial compatibility with Mac OS 9 drivers. Non-driver dependent applications from Mac OS 9 will run in it.
Some people don't like the new interface of Mac OS X. Don't despair if you prefer the older interface as there are utilities that return Mac OS 9's features to Mac OS X. Though many of the utilities need to be kept up to date to continue to be compatible with updates to Mac OS X. So be sure to backup your data before updating Mac OS X to a newer release of Mac OS X if you use any of these utilities:
Apple menu replacements:
Apple menu -> System Preferences -> Classic -> Add Classic menu to menubar
shows Apple menu contents on the Classic menu
Window and icon themes:
Application menu replacements:
Printing Finder Window:
Note: I have not tried each of these utilities. Try at your own risk, backup before using in case one does not work.