(written from a Production point of view)
Unlike the starship designs, shuttlecraft had life-sized full scale mock-up counterparts of the filming studio models for actors to interact with, complete with interiors. The class F shuttlecraft was the first one to be conceived as such. The practice continued with every subsequent Star Trek incarnation including Star Trek: Enterprise and Star Trek, which were otherwise realized entirely in CGI.
The original shuttlecraft as originally designed by Matt Jefferies was to have a more rounded look to it, much like the shuttles of Star Trek: The Next Generation. On his original design Jefferies commented, "Basically it was a teardrop thing, and the whole side panel, the outside door, would slide back, and you could just step right off on the ground. The seats were like bicycle seats mounted on each side of the keel." (Star Trek: The Magazine Volume 1, Issue 12, p. 20) The AMT model company, however, who agreed to build the full-sized set model in their "Custom & Speed Shop", headed by Gene Winfield, as well as the filming miniature for free in exchange for exclusive modeling rights (resulting in their 1974 model kit S595), found that flat panels were easier and cheaper to build. In order to meet their needs, they had industrial designer Thomas Kellogg re-design the shuttle with later contributions from Matt Jefferies who added the Enterprise-style warp nacelles.  "I worked up sketches for it. But AMT, who were going to build the model in their shops in Phoenix in exchange for being able to market the kit of the Enterprise, felt it was beyond their capabilities, so it was designed by Gene Winfield [sic], an automotive designer who had a custom body shop that primarily serviced the automotive industry through AMT. The Galileo as everybody knows it today was not my design. Overall I was a little disappointed, but I think within their capabilities it was a good solution. And it did work, obviously; people did accept it.", Jefferies later remarked. (Star Trek: The Magazine Volume 1, Issue 12, p. 20) Jefferies had to redesign the interior to match the eventual exterior of the studio model. Later shuttle designs by Jefferies after the episode "Galileo Seven", such as a small, two-man shuttle and a bubble-topped space scooter, were deemed either too expensive or simply not plausible with the current special effects of the time and were never used, so the producers stuck with the established design for later appearances of the shuttlecraft. Drawings of all these designs have published in the Star Trek: The Original Series Sketchbook.
Full scale exterior mock-up
The full scale exterior mock-up was a sturdy build. A wooden structure was applied over a basic welded steel framework. The outer skin consisted of hard mahogany press-board called Masonite covered with fiberglass. The curving features on top of the side plates were sheet metal. The center landing gear strut at the end was constructed out of surplus airplane landing gear struts and the nacelles were steel tube assemblies. In addition, a mechanism was built in to semi-automatically open the hatches while simultaneously extending the boarding ramp. "I think we made the windshield panels move up front. Those slid and we made the doors so that they could slide and pop into place on a special track. They were pulled with a cord, like rope on a pulley system, so they did not actually operate electronically.", Winfield explained. (The Ships of Star Trek, p. 102) The mock-up was not equipped with an interior, that being a separately built set at a somewhat larger scale, thereby causing the mock-up being sometimes referred to as the three-quarters scale mock-up. The craft measured 22 feet long, 8 feet high (5 feet in the interior), 13 feet wide (or 24 feet long, 9 feet wide in front, 14 feet in back, and 9 feet high, according to Kiko Auctioneers' listing), and weighed about 2,700 pounds.  Winfield and his team needed close to three months to complete both sets. During its use as a production asset, the mock-up received two paint schemes, the first gloss white for the upper surfaces and battleship gray for the lower surfaces in its original appearance and an overall light gray paint scheme for its appearance as Galileo II.
After production wrapped on The Original Series, the studio donated the exterior mock-up to The Braille Institute (the interior set was demolished after the series wrapped), but the school deemed it an inadequate playing environment for kids and sold it shortly thereafter to a man named Roger Hiseman. After a stint on his front lawn in Palos Verde, the mock-up was moved to an open storage area in Torrance, California where it resided until the middle 1980s. Exposed to the elements, the mock-up deteriorated considerably.  In 1985 the craft was sold to a fan by the name of Stephen Haskins for a reported $1,800, who spent another $8,500 on restoration and had the result displayed at the June 1986 Creation Entertainment's 20th Anniversary Star Trek Convention in Anaheim, California.  Shortly thereafter the mock-up was again moved to an open storage area near San Diego. In November 1986 the mock-up was on display for a week in front of the Palm Desert Town Center theater for the premiere of Star Trek IV: The Voyage Home. During restoration Haskins tried to find a permanent home for the mock-up and even offered it for free to the National Air and Space Museum but was turned down. "We're not into television fiction – that's about it. We are crowded as it is right now, just doing real aviation and aerospace.", then curator Edward Leiser explained. (The San Diego Union, Saturday, June 7, 1986)
In 1989 it was eventually sold to an Ohio fan, Lynn Miller, for a reported $3,000, who had it shipped over to the Canton/Akron airport in April 1991.   There, in cooperation with a local fan club, The Starfleet International chapter USS Lagrange, a second major renovation, done by club members Tim Gillespie, Tim Homa, and William "Buck" Krause, took place under the name "The Galileo Project",   which was also visited by Ed Miarecki.  While still in the process of being restored, the mock-up made an appearance at the LagrangeCon '91 in November 1991 near Cleveland, Ohio, where original builder Winfield was present as well for an autograph session.  Reportedly a disagreement between owner and fan club caused the cooperation to cease during or shortly after 1993 and the owner had the craft moved to an industrial site for again open storage near Akron, Ohio, where it had been spotted until 2008.  The owner of the site lost contact with the owner of the mock-up shortly after it was stored there and when the site went bankrupt in 2008, the mock-up was speculated by some to have been destroyed during clean-up of the site. 
Yet, the mock-up was reaffirmed by its owner to still exist in 2011, as she clarified that, "I originally purchased this to "save" it, but who will save me lol. My Mother always said I had better build a bathroom in it as I would end up living in it. Obviously she disapproved, and as in many things looking back she was right. This thing will either save me or destroy me. After 23 years I am about to sell it one way or the other. Anyone who would buy it look to EBay in the near future. If it does not sell for more than the mimimum [sic] then I guess it will either continue to be stored or maybe a partner will help me finish restoring it. I am hopeful that there is someone out there who has an interest in owning and finishing the restoration. There have been comments about it's destruction or missing parts. The parts are safely stored separate from the main shuttle, as they are restored and I did not want them to be stored outside somewhere."  Being the owner was less of a satisfactory experience as was evidenced in this remark, "I have spent over $100,000 over the years trying to restore and to store the thing. The photo of it in a "Scrapyard" was actually where it was parked undergoing restoration from a guy with whom I had contracted the work. I was called one day and told get it out of here you have one day to do so. I had it hauled to another location where it is in storage." 
The owner eventually put up the mock-up for auction as Lot# 2030 at a local auction house, Kiko Auctioneers, where it was sold for US$61,000 ($70,150 including buyer's premium) on 28 June 2012. The mock-up was acquired by noted Star Trek memorabilia collectors Alec Peters of Propworx, Inc. and Adam Schneider,  whose intent is to have the craft fully restored for the 50th Anniversary of Star Trek.  For this purpose a non-profit corporation, The Galileo Restoration, was founded in order to raise funding, and has secured backing from former Star Trek staffers Doug Drexler and Daren Dochterman.
The physical studio models
Simultaneously with the full scale mock-up, a team of three at Winfield's shop constructed a 22-inch-long filming studio model. As was commonplace in that era, the model was mostly constructed out of wood, with metallic features, such as the landing gear. After completion the model was sent to Linwood G. Dunn's Film Effects of Hollywood effects house where footage was shot of the model flying in space and medium-range footage of the model in the just-completed model of the shuttlebay of the USS Enterprise (for this purpose the model of the shuttlecraft was built in scale with the shuttlebay). The footage shot was used throughout the remainder of the series.
After the series wrapped, sight was lost of the model and it was believed to be lost or to have vanished during the clean-up of the Paramount Pictures lot in late 1973. However in 1987, when pre-production of TNG Season 1 was in full swing, members of the production staff discovered the model on a pile of rubble in a forgotten corner of the studio. Broken in half and missing the forward bulkhead with the windows, the landing gear, and the corrugated wrappers around the rear of each engine pod, the model was refurbished as much as possible and used as set dressing in TNG: "Lonely Among Us". Intended to be a generic model of the class F shuttlecraft, the new paint scheme was not a faithful recreation of how the model originally appeared. The missing bulkhead was replaced by a smoked Plexiglas sheet.  Four years later, fully restored and with a paint scheme corresponding with the original paint scheme of the full-scale mock-up, the model was featured in the 1992-1993 Star Trek Smithsonian Exhibit and was a year later loaned to the Hayden Planetarium, New York City, for its 1993-1994 exhibition. Since then the model has not been seen publicly, but is presumably still in the possession of the studio.
In 1996, a tiny miniature of a class F shuttlecraft, representing the NCC-K7, was constructed by Jason Kaufman at Gregory Jein's workshop for DS9: "Trials and Tribble-ations". On his own initiative Kaufman asked for and received permission to build a detailed shuttlebay on the Deep Space Station K-7 studio model, complete with a shuttle and the Spacematic miniatures. The bay and miniatures were built from scratch with parts and pieces lying around in Jein's shop. (The Magic of Tribbles: The Making of Trials and Tribble-ations, p. 43)
A first CGI version of the class-F shuttlecraft was inserted into the 2001 director's edition of Star Trek: The Motion Picture by Foundation Imaging as a subtle homage to The Original Series, where it was seen taking off from the San Francisco air tram station.