You can access the files on the drive from a newer computer and OS, but the problem with using a newer OS is that some files can only be opened by a particular version of the application that only runs under an older OS version. For example, say you have tax-prep files from year 1998 in OS 9 and those can not be opened by current tax software 2012 under OS X. You can usually import or translate files from one year's version to the next years, but over many years you have lost any backwards compatibility. If you have Excel spreadsheets from version 1998 you may be able to open them with 2010 version of Excel, but maybe not with 100% translation of all features or formatting, hypothetically.
If your G4 was running OS 9 and the files you need can only be opened by running the original version of the software under OS 9, then your best bet would be to search your favorite local classified ads for a similar vintage computer that is in working condition, and use the external hard drive with that computer. If you consider the worst-case that the G4 requires some major repair like a motherboard or power supply. Grab a used working model similar to yours and get back in business, that may be cheaper and faster than trying to figure out what is wrong with this one and fixing it, if you are facing some kind of fiscal-cliff or deadline for accessing your data files.