Well, the first thing I generally check with nonfunctional equipment is the easiest things...external communication (GPIB, serial), ensuring voltages are correct (that 3.3VDC test point is close to 3.3V, same with any other voltages noted on silkscreen/in service manual), unseat and reseat any socketed IC's, see if there are any unpopulated {or populated} footprints that could be looked at (ex. JTAG, serial), sniff around with logic analyzer.
I own a 34401A, although I've never had an issue with it and have not had a reason to become familar with the troubleshooting process of it. If you happen to have a second instrument that you can compare voltages, signals, and responses to, then it can make the whole process much easier/quicker.
Try what you can and let us know what you find and I'm sure some more people will try to help (either I or some of the much more knowledgeable people on this site)
Good luck!