I have one system with 386 processor having hard disk and a few systems with 486 processor having no hard disks. Can I network these systems using the 386 system as the server in DOS? If yes, how can I do that, i.e, hardware, software requirments, and the process?

Computer networks exist long before the advent of Windows. Of course you can: http://huizen.dds.nl/~jacco2/samba/dos.html better yet: http://www.hippy.freeserve.co.uk/msdosnet.htm

Have a look at Novell Netware Lite, together with cheap or free old network cards. For a slave, one floppy has enough space for booting and novell. If one of the 486,s has a higher spec than the 386, would it not be worthwhile using that as the server? Good luck - keep us posted.