Echoing others that this is definitely possible and done more often than people realize. Fixed wing aircraft are more common simply because they are less expensive, generally faster, and have a longer range (plus it's cheaper/easier to learn to fly on one). You could get a decent used gps-equipped 172 for the price of a luxury car, cruises at 120mph, and it costs ~$100 an hour to fly, fuel included.
Helicopters are usually more expensive to purchase and more expensive to maintain and operate, though not too terribly much. Helicopters need a little less space to takeoff/land, though not as little as you would think. A little cessna doesn't need much ground roll to takeoff or land and there are tons of landing strips all over the US in almost every rural community I know, so I doubt this would ever be an issue. Neither this nor fixed wing aircraft will ring in as cheaper or equivalent to driving or flying comercially, but it also isn't unaffordable and depending on the value of your time may actually be worth it. It becomes an economic no-brainer if you have a couple partners who also fly and you planepool together. I know some guys in construction who do this regularly when they have a job out of town.
Biggest drawback to doing this is weather and maintenance. You would definitely need an IFR ticket if you want any reliability (so you can fly in poor weather, low visability conditions), and even then certain weather conditions could ground you just like they do the big airlines. Maintenance issues can also ground you, as can all the required inspections so these need to be planned well in advance.
Overall I think that general aviation is a lot of fun and can often be very useful. I feel that many people aren't aware of just how affordable and doable it is and I would encourage anyone to check out their local airport and take an intro lesson. Most places will give you a great deal on a first lesson which includes ~30 minutes of flight time!