You can back up the data a number of ways... Easiest if you are not good with hardware is to Laplink it to another system in which you copy the drive entireties to another systems larger drive. There was a FX program a while back as well as Filevan that did this from Rainy City Software, but they are all not free. You can send data via LPT (Parallel) or Serial port from one system to the next to back it up and migrate data to new systems.
Or you can install a second hard drive in the case as a slave drive and XCOPY the root and all data to the other drive.
As far as Windows and DOS, Windows 3.x, 95, and 98SE are DOS friendly. I would avoid Windows Me. You can also run a newer OS if the hardware supports it such as if your system is at least a 200Mhz Pentium 1 CPU with 64MB Ram and a 2.5GB HD you can install Windows 2000 Pro. And if your system is at least a Pentium II 233Mhz with 128MB Ram and 8.4GB Hard Drive you can install Windows XP Pro SP2. Windows 98 SE and all prior Windows OS will run fine on less than 200Mhz CPUs and less than 64MB Ram.
If a 286, 386, or 486 or better CPU, and 1MB+ Ram, you can run Windows 3.11
If a 386 or 486 or better CPU, and 8MB+ Ram you can run Windows 95
If a 486 or Pentium CPU, and 16MB+ Ram you can run Windows 98 SE
If it is so old that it is an 8088XT CPU, you could try to find Windows 2.0 and run that, but you will need 512k Ram. Pointless unless you had an old app that had to run in a Windows OS that was supported by Win 2.0.
Also to note, your DOS version will change from 6.22 to 7.xx if you install Win 95 or 98 or newer, Windows 3.11 will stay with 6.22 and you can use the memmaker etc to get EMS etc set teh way you want and keep your base 640k as close to 640k as possible to keep the performance up, and avoid problems with too low of base memory that some apps will puke about if you are running in the 500-560k base memory range or less due to hogged base memory resources of a ugly config.sys and autoexec.bat setup.