How accurately have we determined the fundamental constants of the Standard Model? Less accurately than you might expect.
The data from the particle data group as of 2011 sets out the story.
For quark masses, the range of values within the appropriate confidence interval (generally 95%) is as follows (the most likely value is not necessarily in the middle of the range since some confidence intervals are lopsided):
up quark = 1.7-3.1 MeV
down quark = 4.1-5.7 MeV
up/down quark mass ratio = 0.35-0.60
mean up and down quark mass = 3.0-4.8 MeV
strange quark = 80-130 MeV (central value 100 MeV).
strange quark mass/mean up and down quark mass = 22-30
charm quark mass = 1.18-1.34 GeV
bottom quark mass (by MS definition) = 4.13-4.27 GeV
bottom quark mass (by 1S definition) = 4.61-4.85 GeV
top quark mass = 171.4-174.4 GeV
The strong force coupling constant related to interactions of quarks and gluons with each other and used in quantum chromodynamics (QCD) is known to an accuracy of approximately 0.6%, which at the scale of Z boson mass is 0.1184 ± 0.0007, although the range of values reported from a robust variety of methods suggests an actual value of closer to 0.119 +/- 0.001.
The canonical value of gluon mass in theory is zero, but a gluon "mass as large as a few MeV may not be precluded" by experimental data.
The first principles estimate of proton and neutron masses from QCD, for example, from theory and the known constants, is accurate to only within about 5%. Given the limited accuracy with which the constants upon which these estimates are based, however, this is quite impressive.
For fundamental leptons and fundamental bosons and quantum electrodynamics (QED) we have much more accurate estimates, although our absolute neutrino mass estimates are far less certain:
electron 0.510998910 ± 0.000000013 MeV
muon 105.658367 ± 0.000004 MeV
tau 1776.82 ± 0.16 MeV
Koide's formula for charged leptons which states that the square of the sum of the square root of the three lepton masses, divided by the sum of the lepton masses equals 1.5, is similarly precisely established.
electron neutrinos < 2 eV (probably closer to 10^-5 eV)
The difference between the electron neutrino and muon neutrino mass is on the order of 10^-5 eV and the difference between teh tau neutrino and electron and muon neutrino masses is on the order of 10^-3 eV. But, both the mixing matrixes and masses of neutrinos are not known with great accuracy.
W = 80.399 ± 0.023 GeV
Z = 91.1876 ± 0.0021 GeV
Z-W = 10.4 ± 1.6 GeV
The fine structure constant, which is the coupling constant of electromagnetism is α = 1/137.035999084(51).
There are additional constants in the Standard Model, of course, principally, the weak force coupling constant, the four parameters necessary to determine the 3x3 unitary CKM matrix (which can be parameterized in more than one equivalent way), and the four parameters necessary to determine the 3x3 unitary PMNS matrix, but those are less familiar. The first two are known with precision similar to that of the other weak force observable, while the PMNS matrix elements related to neutrino oscillation are known to accuracies only within roughly a factor of two for values close to zero (or with an 1-value factor of two for numbers close to 1).
There are experimental constraints on the minimum masses of hypothetical fourth generation particles, higher generation W and Z bosons, Higgs boson masses, and masses of other hypothetical particles under various extensions of the Standard Model and for a variety of interactions and decay modes prohibited by the Standard Model (e.g. leptoquarks and axions), but, these are largely consistent with zero although recent neutrino oscillation data most strongly supports more than three generations of neutrinos. Some of the most theoretically well motivated extensions of the Standard Model (predicting proton decay, neutrinoless double beta decay, magnetic monopoles, etc.) turn out to have some of the most experimentally prohibitive boundaries on their existence.
One of the implications of this is that there is very little freedom experimentally for innovation theoretically in electroweak interactions, but some meaningful freedom experimentally for adjustments to quark masses or even to QCD equations.
By comparison, the uncertainty in the gravitational constant's measurement is about one part per thousand, the uncertainty in the cosmological constant's value is about 3.3%, the experimental uncertainty in the measurement of the geodetic effect of general relativity is about 0.2%, and the experimental uncertainty in the measurement of frame dragging in the vicinity of the Earth is about 19% (all of which are consistent with general relativity's predictions). The geodetic and frame dragging effects are measured in terms of deviation from the Newtonian gravitational prediction.
The speed of light in a vaccum, necessary in both quantum mechanics and general relativity, is known with such precision that we use it to define the length of the meter relative to the length of the second.
It is also worth noting that it is not possible to do any remotely complicated calculations in QCD, or very exact calculations in general relativity with a number of bodies on the order of the number of stars in a typical galaxy given current computational technologies and mathematical methods. These kinds of calculations are possible in simple, stylized systems, but not in systems even remotely approaching the complexity of those seen in real life.