You can do this using Partition Magic and Smart Boot Manager.
Partition Magic is commercial software. Smart Boot Manager is GPL Free Software.
You might be able to get the whole thing done using Free Software entirely. Partition Resizer and FIPS come to mind. But I suggest Partition Magic for the GUI, ease of use, and amount of control.
Start up Partition Magic, resize (shrink) your Win 98 partition. DOS has to be installed within the first 4 gigs of the HD, so set your options accordingly. When you shrink your Win 98 partition, you will have some space left on the HD. Create a FAT16 partition there. Finally, format the FAT16 partition.
Partition Magic will queue up all these operations without performing any of them. Once you're done instructing Partition Magic, click the green button, and it will go through the motions and do everything you told it to. It may have to reboot once.
Once you're done with Partition Magic, install Smart Boot Manager. It comes with HTML docs, so you should be able to figure out how it works. Once you install it, and reboot, it will give you a list of all of your partitions, and ask you which one you want to boot from. You choose the FAT16 partition and press enter.
Of course, at this point, there is no OS on your FAT16 partition. So your PC will panic and will say something like COMMAND.COM NOT FOUND PLEASE INSERT SYSTEM DISK AND PRESS ENTER. This is when you insert MS-DOS 6.22 floppy #1 and I assume you can figure out the rest.
Good luck.