There are many options as they already have said.
Disconnecting HDDs and connecting them to your modern rig is still an option even for a Mac PC. This is 0 investment method normally, but in your case a bit of a purchase might be required. There are SATA/IDE HDD docking stations around with USB/eSATA interfaces. Just as an example:
http://www.ebay.com/itm/Dual-2-5-3-5-SATA-IDE … =item25907e071d
The other 0 investment method is burning CD/DVDs as required, but it's tedious just like FDDs and ZIP drives. Yes, not that much, but close.
If you have working USB ports in your retro rig, your retro OS is something like Windows 98 or newer (windows 95 OSR2.1 or 2.5 are also somewhat OK, but not as less problematic as W98), and you don't care about pure DOS data transfer, you can just easily use your USB sticks, providing that you have Windows 98 drivers for them on hand. Sandisk Cruzer Micro USB sticks with 2 and 4 GB capacity that I have, are working just fine with their official W98SE drivers, according to my experience. There are also unofficial but very popular universal USB storage drivers for Windows 98 around.
You can also try one of the USB/floppy converters, but be ready to be a problem solver, if you want the cheap ones (i.e.: Gotek). They are either expensive and less problematic, or cheap and relatively problematic (too many versions to chose from, shitty software, etc.).
Network transfer is also another option, as long as you accept to go thru all the hooplas required to connect two devices using two different OSes from different ages trough the same modern network.
Another option is using Compact Flash Cards or SD cards with bracket or drive bay type CF/IDE or SD/IDE interfaces. Something like these:
http://www.ebay.com/itm/KDQ4-CF-to-IDE-3-5-Ma … =item4d142838dc
http://www.ebay.com/itm/Brand-New-3-5-40-Pin- … =item2c943ed3b6
or even:
http://www.ebay.com/itm/STARTECH-COM-35BAYCF2 … =item418ee32cd8
This method is my preference for pure DOS data transfer, and believe me, I tried all of the above, in the past. 😢
GA-6VTXE PIII 1.4+512MB
Geforce4 Ti 4200 64MB
Diamond Monster 3D 12MB SLI
SB AWE64 PNP+32MB
120GB IDE Samsung/80GB IDE Seagate/146GB SCSI Compaq/73GB SCSI IBM
Adaptec AHA29160
3com 3C905B-TX
Gotek+CF Reader
MSDOS 6.22+Win 3.11/95 OSR2.1/98SE/ME/2000