There is no doubt that the FAA envisions flight under Instrument Flight Rules in Class G airspace as something that may be done in compliance with the regulations.
For example, as another answer pointed out, page 4-3-2 of FAA Order JO 7110.65W gives an example of an IFR clearance containing the following language:
WHEN ENTERING CONTROLLED AIRSPACE (instruction), FLY HEADING (degrees)
UNTIL REACHING (altitude, point, or fix) BEFORE PROCEEDING ON COURSE.
Also, as another answer noted,
Most non-towered airports are Class G until 700 or 1200 AGL. There are
MANY non-towered airports with IFR approaches and ODPs. Every time you
descend while IMC into Class G on a published approach or depart on an
ODP or Diverse Departure after being released for an IFR departure,
you are legally flying in Class G while IMC.
Here are examples of approach and departure procedures published for an airport in full-time Class G airspace (Class E airspace begins at 700' AGL in this location) -- https://flightaware.com/resources/airport/KCVO/procedures
Another answer mentions a 1993 NTSB ruling finding that a pilot who took off in uncontrolled airspace in IMC conditions with no IFR clearance at all was in violation of FAR 91.13.
Page 5 of the same ruling states the following:
According to FAA aviation safety inspector Lawrence Smith, the
standard procedure for taking off from an uncontrolled airport in IMC
is to seek an ATC time-limited clearance to depart from the airport
and fly into controlled airspace according to a pre-filed flight plan.
(italicization and bolding added)
Therefore the answer to the actual original question
Let's say an IFR flight was planned to depart from a non-towered
airport with class G airspace extending from the surface to 700 feet
AGL. The flight visibility and ceiling both go down to 0. Can the
pilot get an IFR clearance?
is unambiguously "yes", at least if the pilot is operating under part 91. (For why this matters, see https://www.thinkaviation.net/standard-takeoff-minimums/ .)
If the original question were "Do the FAA and NTSB look favorably on flights conducted in IMC conditions in Class G airspace with no associated IFR clearance of any kind?", then the answer would have to be "no".
It does seem unfortunate that this latter point is not explicitly spelled out in the regulations. Likewise the issue of what actually is the distinguishing characteristic of a flight that is operating under "Instrument Flight Rules"-- is it simply the fact that an IFR clearance has been issued in association with the flight, or is it something else? These are apparently matters that have somehow "slipped through the cracks", just like the issue of clearly stating whether various regulations prohibiting certain activities in Class-E-to-surface airspace surrounding an airport also pertain to the airspace enclosed by a Class-E-to-surface "extension". (For example see Does FAR 91.155c apply to class E surface extensions? )