TIPI provides a hierarchical file system (essentially a hard disk as far as the TI is concerned) for more information see the TIPI description here or http://ti994a.cwfk.net/TIPI.html
Myarc produced a Winchester Personality Card that allowed you to connect a WDS-100 SASI controller to your TI and then some MFM hard drives to the controller. This supported up to two 20meg drives. There are only three I know of in collectors hands.. Good luck on finding one!
Myarc produced a HFDC controller that in addition to disk drives supported up to 2 MFM hard drives as well, I have a 40mb one on my Geneve, not sure the max limit.. (rare)
S.N.U.G produced an excellent SCSI controller card that is compatible with the TI and Geneve. These are difficult and expensive to find for sale. http://www.s-n-u-g.de/home/index_e.php
WHSCSI card (Western Horizon SCSI card) Handles 7 SCSI drives and with a HP SCSI port splitter box could handle 6 more SCSI drives for a total of 13 SCSI drives (rare)
Therry designed a IDE controller for the 99/4a and some have been built as part of a group project among TI enthusiasts. These were recently produced by shift838 and available below, they are open source so you can build your own if you like surface mount soldering.