Yes, you can watch the lunar eclipse, no problem. It is a celestial phenomenon. Only solar eclipse should not be watched with naked eyes. You should wear some glasses because the rays are harmful.
Why our ancient people said not to eat at the time of eclipse and before eclipse is so that the food is all well digested and you can do meditation. During eclipse time when you do chanting, meditation, or pray, it has a hundred times more effect - this is what is said.
The sun, moon and earth are in one line and so the cosmic rays are flowing such that at that time any sadhana you do has manifold effect.
So during the eclipse do your meditation, chant ‘Om Namah Shivaya’ or ‘Om Namo Narayana’.
If you chant 108 times during an eclipse, that means you have done it ten thousand times.
It is a very auspicious time for sadhaks; it is not a bad time. It is a bad time for enjoyment and for pleasures.