That computer may be limited to using hard drives smaller than 127 GB ... I think another threshold was 512 GB.
The safest best would be to buy one or a couple of cheap 60-120 GB SSD drives. They should work just fine and transparently with an ATA to SATA adapter like this one:
https://www.newegg.com/Product/Product.aspx?Item=N82E16812200156These adapters (the chip on the adapter) probably won't support ssd features like TRIM but as you're not creating a lot of files, the SSD won't need to shuffle data around anyways so there won't need to do TRIM and all that would normally be required on a SSD with high usage, therefore it's not really a big issue.
Yeah, like Ian.M says, you'd probably have to create partitions and make them under something like 32 GB (if you use FAT32 for Windows 95/98) or I think under 8 GB for FAT16
I'd just make a disk image of your whole drive for backup purposes.
Other than the hard drive, i see as a concern just the power supply. May want to open it up and maybe replace the fan with a new high quality one, and while you're there check the existing capacitors to see if they're still within acceptable parameters.
Check youtube channels like Phil's Computer Lab
https://www.youtube.com/channel/UCj9IJ2QvygoBJKSOnUgXIRAHe does a lot of vintage pc, dos / windows videos, and he has some videos with ms-dos software and utilities for accessing internet and transferring files over network from dos, i remember seeing a video on his channel about internet/network stuff ... let me see if i can find it ...
Yeah, here's one of them :