An old IDE drive taken out of regular use has a significant chance of failing due to stiction just sitting on the shelf. A 250MB IDE drive and MSDOS 6.2 would date the system to the mid '90's, so that drive has been running whenever the machine has for the last 25 years. . .
IMHO it would be foolish to trust it as the only backup, and as I mentioned earlier, I'd get as much data off it as possible in-situ. Even if you have to resort to removing and imaging it, having a copy of the filesystem will be useful if the image turns out to be partially corrupt.
Once you have the data, either a filesystem copy, or a disk image, that can be archived and backed up as many times as you need for security, together with the tools and information needed to reconstruct a bootable drive from it.
If you have to image it, take a first shot at it with something dumb like DD, that won't force it to retry multiple times, homing and re-seeking to try to read corrupted data, as if the drive is on its last legs, that sort of intensive activity can kill it.
Imaging in-situ and non-invasively to a file on the hard drive of a second PC is possible if you've got an imaging utility that runs under plain MSDOS, the original system can still be booted from floppy and you make a DOS boot floppy that uses INTERSVR/INTERLNK to mount a volume on the second PC on the original PC for the imaging utility to save to. Don't expect it to be fast - 80KB/sec is on the high side for a standard parallel port connection, so it could easily take a couple of hours for a 250MB drive.
If you do an insitu filesystem copy, DEBUG (which should be present if it has a 'vanilla' MSDOS 6.x installation) can be used to grab the MBR*, boot sector and partition table, and stuff them in a file for later use if you need to manually patch together a bootable image, but if you've identified the DOS version correctly, you shouldn't need that as its easy enough to use a VM to recreate a bootable drive from an image of the original MSDOS system disks, using FDISK and FORMAT /S then copy the filesystem contents onto it. Depending on the VM software, it may well be possible to mount the raw CF card in an IDE adapter on a USB to PATA adapter as IDE drive 0 of the VM.
However if the old PC BIOS has geometry limitations, it may not boot from a CF card written on a more modern PC and you may have to make a real MSDOS boot floppy with FDISK, FORMAT and INTERSVR on it, and format the CF card on the old PC to make it bootable on the old PC then copy the filesystem back onto it using INTERSVR and INTERLNK + your preferred file management utility.
Its not my first time at this rodeo, and I've had to cope with getting data off a system with no usable floppy drive (and no way to fit one) or CD drive or bootable USB capability, and no way to read the hard drive on another system. Hint: Laplink 3 is your friend as long as it boots DOS and has a working COM port, as it can be installed onto a DOS system over a serial link!
* Reading outside DOS visible volumes (i.e. with a drive letter) with DEBUG e.g to backup the MBR, is only possible by keying in an assembly language routine to do a raw disk read to RAM and executing it from DEBUG. Therefore if the hard drive is reasonably healthy or the floppy drive works, you may want to use a better disk sector editor utility that can handle raw disk access.