The cited paper, and ones like it, are young earth creationist nonsense. The Geoscience Research Institute is an arm of the Seventh Day Adventists, whose official position is that "in a recent six-day creation the Lord made “the heavens and the earth, the sea, and all that is in them” and rested on the seventh day." This is a central tenet of Seventh Day Adventism (it's in the very name of the religion); teaching or writing about an Earth that is billions of years old is explicitly forbidden.
Young earth creationists are wont to attack radiocarbon dating because it has significant limitations. Samples can be contaminated in situ, during extraction, during storage, in transportation, and in processing. Contrary to Giem's claim that "the hypothesis that machine background can account for this carbon-14 has been universally rejected by researchers in the field, and for good reason," several of the papers cited by Giem were scientists trying to assess the machine background and other limitations of their techniques.
For example, Giem cited Aerts-Bijma et al. seven times. This paper looked at "AMS sample handling in Groningen." This lab attempts to perform radiocarbon dating in an automated, assembly line fashion, taking about an hour to perform an analysis. This is fine for archeology studies with budget constraints. The paper by Aerts-Bijma found their approach has limits.
One issue with carbon dating is that the previous run can contaminate the next. To test this, Aerts-Bijma et al. ran two sets of tests against the same sample of anthracite. In one, they interspersed each measurement by carbon dating 100% modern carbon. This gave an artificially high carbon-14 reading for the anthracite. In the other, they didn't do such a dumb thing and found a lower floor. Giem reported the intentionally mishandled reading.
Other labs have lower floors by following more stringent but significantly more expensive procedures. Carbon compound scintillators such as those used to detect neutrinos need carbon that has extremely low amounts of carbon-14. The best source of such carbon is old hydrocarbons that have not suffered in situ contamination. Techniques have been developed to measure carbon-14 to one part per 1018 to ensure that those sources are indeed old and uncontaminated. These techniques are well beyond the budget of a typical archeological study. Giem did not report these because they go against his claims.