OS/2 is not open source, nor is it free of charge. However you can get it on EBAY for really, really cheap amounts of money. I've seen it for like $3-$10. If you go that route, you should go with version 2 or Warp 3. Version 1 was weak, and Warp 4 really needs 8 MB.
Also, you didn't mention if it had a CD-ROM. If it doesn't, you should get the diskette version of OS/2.
There's lots of free software for OS/2 on Hobbs.
You can get DR-DOS free of charge for personal use still I think. Other DOS (MS-DOS, PC-DOS) would work too. FreeDOS is available, but not sure how stable it is at this point, and you may run into software compatibility issues.
OpenGEM and FreeGEM are open source and free of charge.