Yes, this is completely possible. In fact, I keep a bootable USB of SLAX(
www.slax.org) with backups of important files with me just in case of a hard drive failure. Go to that website, and download the tar file. Extract the contents to a pendrive and if you're on windows, open command prompt and go to the drive letter of the USB, change directory into "boot" and run bootinst.bat (this process can be done by double clicking on the .bat on windows, I think), or if you're on Linux, open terminal and go to the place where the USB is mounted, cd into "boot" and run the file bootinst.sh as root ("sudo sh bootinst.sh" if you're running Ubuntu)
I don't have a BSNL connection, so I'm not sure about this, but they should have an external router capable of dialing into BSNL. If you set it up so that the router dials into the internet connection instead of having to dial through the computer, then you just need to connect the router to the computer through ethernet (LAN port) and then turn them both (router and PC) on.