How do wide-bodied commercial aircraft achieve pressure equivalent to 6000 feet altitude?
The killer in fuselage pressurisation is the cycling of the structural loads. Aluminium is very unforgiving of repeated load changes: While steel has a load limit which can be applied infinite times, no such limit exists for aluminium. This means an aluminium aircraft will eventually fail if it flies enough cycles. It also means the plane can be built lighter or operated for more cycles if the pressurisation loads are lower.



Fatigue curves of steel and aluminium (By Andrew Dressel at English Wikipedia, CC BY-SA 3.0, source)

The consequences of this nasty characteristic were illustrated in Aloha Airlines Flight 243 in 1988, when the upper part of the forward fuselage failed at the suction peak aft of the cockpit and more than 5 meters of fuselage skin were ripped away.



Aloha Airlines Flight 243 after landing (picture source)

A fuselage made from carbon fiber reinforced epoxy shows very little fatigue and is much more tolerant of repeated loading changes. Here, thermal and impact loads are the most critical issues. However, the bad historic experience with aluminium structures means that not only the engineers, but even more the certification authorities are very careful to clear any new material for high, repeated loads. As a consequence, the graphite fuselage structure of the Boeing 787 is much stronger than a regular fuselage made from aluminium and can be loaded to a higher pressure without much added risk.

Whether this means less headaches and constrictions for you is less clear. The added pressure does feel better, but the major improvement in creature comforts in the cabin of the most modern airliners results from  higher humidity in the cabin air.

Most single-engined aircraft have no pressurisation; only the high end types like the Piper Malibu or the TBM 700 do. On the other hand, some business jets can keep the cabin at sea level pressure up to 12000 m altitude. In airliners, however, holding the cabin pressure at or above the equivalent pressure at 8000 ft of altitude is mandatory. Citing Wikipedia:


  Keeping the cabin altitude below 8,000 ft (2,400 m) generally prevents
  significant hypoxia, altitude sickness, decompression sickness, and
  barotrauma. Federal Aviation Administration (FAA)
  regulations in the U.S. mandate that under normal operating
  conditions, the cabin altitude may not exceed this limit at the
  maximum operating altitude of the aircraft.