It means that the old man has chosen not to take sides in the Spanish Civil War.
This would make him a rare individual indeed, because virtually all Spaniards supported one side or the other, Republican or Nationalist. This bitter, bloody conflict arose out of profoundly different world-views, and most people stuck tenaciously to those world-views right throughout the war, which helps to explain why the Spanish Civil War lasted as long as it did.
The old man, however, is an exception. He doesn't appear to have any deeply-held political convictions. But ultimately, this doesn't make any difference; he's every bit as vulnerable to the horrors of war as the most fanatical Communists or Fascists. That's one of the story's overriding messages: that war effects everyone, including those who are "without politics."