A lot harder than one running on e.g. ArchLinuxArm.
The root of the problem is installing calibre's dependencies, and the lack of a package manager. AIUI, OpenELEC is stripped down to run very efficiently with just Kodi, and you aren't *expected* to install other things.
You'd have to manually compile all the dependencies by hand, using "./configure; make; sudo make install" and friends, struggle with the read-only system image, etc. -- OpenELEC really doesn't want you messing around with the system!
You could get a stripped down system that very efficiently runs only what you need it to, if you go the Gentoo route -- ArchLinux uses a (heavier) generic base system -- and also benefit from a package manager.
ArchLinuxArm or Gentoo, either way will require more effort to set up.
Or use the debian-based XBian or OSMC, and pull their (old?) calibre packages.