Yes, but there is NOTHING in between Hawaii and the west coast of the US. Not a single place to put down even a single-engine airplane, much less an airliner. Call it a 340-minute trip, and at the midway (Equal Time) point you are 170 minutes from the nearest airport - not 207 minutes, but getting close to it (and only 10 minutes away from the ETOPS rule prior to the 777).
There are several places in the mid and north Pacific that can be used as emergency divert fields, and thus a trip from Hawaii to Guam might never have you more than 130 minutes from a suitable airfield. (WAG on my part - haven’t flown across the Pacific in several years).
Several factors are considered when airlines design their long-haul routes, and emergency divert fields are definitely high on the list. As 1920s Style “Death Ray” pointed out the ETOPS rules restrict twin-engine airplanes, and for Pacific or Polar routes they may fly non-great-circle routes in order to stay “close” to divert airfields. It may add a couple of hundred miles to the overall flight distance but it keeps a viable alternate available.
Go to that Great Circle site that Rick linked to and type in LHR-NRT (London Heathrow to Narita, Tokyo). Not exactly what you might expect. For an even more dramatic one, American Airlines flies from Chicago O’Hare to Delhi, India. (ORD-DEL). The route goes very near the North Pole, but AA flies this route with a 777 (a twin-engined aircraft) so the actual route takes some zigs and zags to keep an airport within 207 minutes at all times.
Aircraft with more than 2 engines (MD-11, 747, A340) can ignore the ETOPS rules and just fly great circle routes optimized for wind. Still, you’d be hard pressed to find a route that gets you much more than 240 minutes away from an airport unless you start constructing make-believe routes (such as flights to and from McMurdo Station in Antarctica). The one from Sydney to anywhere in South America is a good one as well. Still, 240 minutes is 4 HOURS and while the ETOPS rules are designed to mitigate danger due to engine failure, other problems would leave you in a lot of trouble. A Swissair MD-11 was lost off the east coast of Canada after an electrical fire - the time from the first noticeable smoke until they went into the water was 11 minutes.
Back to the OP (or someone who mentioned it) - I never saw the episode of House but the EWR-SIN route is the longest scheduled airline route now, so that’s probably why they picked that flight. Normal scheduled time is somewhere around 18 hours to the US and almost 20 on the way back. Singapore Airlines uses an A340-500 (a 4-engined aircraft) on the route, so ETOPS rules are not a factor. The flight takes a Polar route, so they could be quite far from an airport, but (guess here) not much more than 240 minutes.
As an aside, if a passenger is sick and urgently in need of medical care a flight will normally divert (I’ve done it quite a few times). If a passenger dies, policy is to press on to the destination.
