I suppose for a single individual, a bike plus a good bike trailer (or maybe a backpack and/or large panniers) would be a good option. I have a wife and four kids, though... oh, and I don't live in an urban area.
I'll stick with my SUV + camp trailer, thanks. The trailer is always packed with canned food, cooking & heating fuel, water purification equipment, has solar panels for electricity, etc. It's also got tents, sleeping bags, and all sorts of other assorted camping equipment and tools -- including the stuff we'd need if we decided to abandon the vehicle and start walking. As for the guy with the gun, well, I am the guy with the guns :-)
Also, I have a couple of neighbors who are similarly-equipped, and we've discussed how we'd pool our resources in the event of a zombie uprising, caravaning for mutual support.
In an actual disaster, of course, my first choice is to hunker down at home. It's fairly defensible and I have a roughly one-year supply of food stored in the basement. I also have a generator and generally on the order of 60 gallons of fuel in the tanks of my cars and boat (the amount in the cars varies, obviously, but the boat has a 35-gallon tank which is always full). But in the event that staying home isn't a good idea, I can have the trailer hooked up and be on the road in 5 minutes. Given 30 minutes I'll make sure the water tank is full and top up the SUV tank plus some gas cans with the contents of the boat's tank, assuming I can't stop at the gas station. Given a couple of hours I'll coordinate with the neighbors and we'll hit the road together, with everything we can jointly muster.
For other situations, I have 72-hour kits in small backpacks in the garage, one for each member of the family which we can grab at a moment's notice.
I certainly agree about the value of fitness, of course. And I like bikes -- I ride my road bike ~100 miles per week, almost year-round -- but I don't see bikes as a viable way to transport myself and my family.