Specifically, one solution is the resume that lists "key accomplishments" on the first page, following a "summary statement" that highlights your skills. On the second page, under the heading "Employment Highlights (1982-present)", for example, list the most important jobs (no dates), with brief job descriptions - all pointing to those skills and experiences you want to emphasize. That's one, out of many ways, of dealing with this problem.
But suppose you've filled out a job application, which demands exact dates of employment. The interviewer, then, asks specifically about the now exposed gaps. You should have in mind a brief, non-defensive explanation. Your response could be, for example, "There was a family health issue that I had to deal with, and once the situation was over, I was able to start my job search." Very matter-of-fact. No details necessary. (No perspiring or anxiety tremors either!)
What if the gap was the actual job search itself? You could deal with the situation by saying something along the lines of, "I decided, when I left my last job, that I was going to take whatever time was necessary to conduct an intelligent, intensive search. This has included careful research, on-going networking, and evaluation of many options. I think that the process has worked very well..." So, instead of sounding like you've been desperately "looking for a job," you create the impression of a carefully conceived, well thought out job search.
It's important that you don't let the "gap" affect the way you feel about marketing yourself. Everybody has problems in their presentations - these issues can all be addressed satisfactorily, and do not necessarily have to be liabilities.
Resumes do not have to be an exhaustive, detailed history of everything you've ever done. They are marketing tools, highlighting key employment, significant skills, and credentials. A resume does not necessarily have to include every month or every year, nor does it have to even show those troublesome gaps at all.