My leisure travel is mainly alone. I've eaten a lot alone, and usually it doesn't bother me. I have never experienced that the restaurant staff somehow feels akward dealing with me dining alone. I normaly just ask for "table for one". I'm fine eating at the bar and sometimes prefer it.
I usually bring a book to read in. Either a novel or the guide book for the city I'm visiting. Then I can use the dinner to plan for what to see/to do the next day and look at maps to plan the best route/order to visit the places I want to see.