First, let me preface my response by saying "0" sunshine minutes does not mean it was necessarily "inclement weather", nor was it continuously foggy, continuously cloudy, continuously snowing, continuously blowing snow, etc. Looking at the weather records paralleling the sunshine minutes showed several days of relatively nice weather apart from the exclusion of sunshine (ie, low winds, high visibility, mild temperatures, etc). Additionally, during our longest streak, there were periods of clearing during the overnight hours and partial clearing during the day (except where the sun was shining); so while the days may have lacked measurable sunshine minutes, the skies weren't 100% obscured the entire time. So, saying no sunshine means bad weather is like saying A=Z without factoring in B through Y to get there.
So, with all that being said and looking at our weather records dating back to 1932, the longest streak of "0" sunshine minutes was from May 13, 1948 05:55 to May 25, 1948 14:55. That equates to 0 out of 206 available hours (plus a few minutes at the start/end) without sunshine. Our second longest streak was 10/4/1988 05:55 to 10/14/88 14:55; which equates to 0 out of 140 available hours (plus a few minutes at the start/end) without sunshine. The third, fourth , etc. (longest), started landing in the century range and weren't worth reporting.
Staff Meteorologist/Night Observer, KMWN (Mt Washington Obs., NH)