There are three basic ways:
a) Get a LAN card and setup a workgroup with all drives shared and access whatever needs to be backuped to your new machine. This is quite dependant on installed software. But chances are good that some NE2000 compatible card driver is already installed. You have to check what I/O bus system is used and buy the apropriate card. As new PCI or ISA cards are geting rare, Epay should be a good source.
b) remove the disk(s) and attach them via USB adaptor to a new machine. With such you may read the drive from next to any moderen machine, as USB, as well as drives over USB, is quite standardized. This will also require way less hardware knowledge than puting the old drive into a new machine. If they (and I assume there are multiple drives, as 5.7 wasn't the original disk size) are PATA (also called IDE) drives, a $15 PATA to USB adaptor will do the trick. For SCSI drives an adaptor might be less common and way more expensive. Adaptec had once one available, called USBXchange. A Google search might be helpful.
c) if these are USB connectos, all the way to the left, right beside the audio connector, you might be able to just plug a USB drive there. I don't remember if Win98 supported USB disks out of the box. Have you tried to plug in a USB stick? Just try it. Preferably one with less than 32GiB and FAT32.
For installing any drivers (or other software), and if you got no other machines with floppies, just use the CD. It sould be able to read writeables. Make sure to burn a CD not a DVD or Bluray.
This brings us to way #4:
d) Use some kinde of old fashioned file transfer programm. Put whatever programm you prefer on a floppy (or CD) and run it on both machines (the old one and some new one), after connecting them via a serial cable. Some USB to Serial if your new machine doesn't offer any classic serial port. Cost is about $15 for the USB/serial. and maybe another $10 for a gender changer.
Oh,and there's one more, without the need for any hardware tinkering:
e) Boot a live Linux from CD. Then USB will then most likely work and you may use whatever USB stick to transfer your disk data within short time. Try Knoppix, as it's well known to come up even on real weired machines. There are many other small, direct bootable Linux distributions, but Knoppix is well known for easy handling even if you're not a Linux buff. Just make sure you pick a CD-based one.
(As more as I look at this machine, as less it seams like a Retrocomputing case)