I'm looking to get rigorous definitions on which to base the important quantities in classical mechanics. To me a "rigorous" physical definition is an operational definition -- that is one in which we define how to measure the quantity over the course of an experiment -- or a definition in terms of other quantities which are operationally defined.
Time can be measured (at least in classical theory) arbitrarily well with clocks. Whether or not we can actually build a clock with the necessary accuracy is not important for this definition as long as we can conceive of one being built.
Position can be measured arbitrarily well with measuring sticks.
Using these two quantities above we could conceive of strategically placing an array of measuring sticks and clocks (that is building a reference frame from them) so as to entirely measure the motion -- position, velocity, acceleration, etc as a function of time -- of any object we care to.
Now I turn my attention to the most important equation in Newtonian mechanics: $F=ma$ (or $F=\dot p$ it doesn't really matter in this case). For this equation to make any physical sense we need to be able to measure the mass and/ or force (and/ or momentum) of an object during any experiment arbitrarily well.
How can we define force or mass (or momentum) in such a way that in the midst of some complicated motion -- possibly involving losing/gaining mass, etc -- we can still be certain that we can measure values for these quantities?
One note on this: I would prefer not to assume the equivalence principle as a way of measuring mass. Let's just pretend that inertia mass and gravitational mass are not the same because, as far as I can tell, in classical mechanics there is no strong reason to believe the equivalence principle holds based solely on Newton's laws (and other equally important "first principles" like the conservation laws). Of course, there's empirical evidence, but again let's just ignore this for now and see if we can find another way of measuring mass (or force or momentum or kinetic energy or any other thing which as allow us to obtain a measurement of mass indirectly).
EDIT: This will be a response to the question that Qmechanic links to (admittedly a similar question) detailing why I find the answers therein unsatisfactory. My hope is that either someone can give a new answer or can assuage my uneasiness about one or more of those answers to the linked question.
The top answer is by joshphysics.
The part about forces (and masses) is his statement of the third law: " If any two objects are being observed in a local inertial frame, then their accelerations will be opposite in direction, and the ratio of their accelerations will be constant." Here he implicitly assumes that two objects in an inertial frame will be accelerating. So I think he just forgot to add that there must also be so interaction (force) between the two objects.
If we assume that there is an interaction between the two objects then this is just Newton's regular old third law (pretty much). However, this definition only provides a means of measuring mass if the accelerations of the two bodies are completely due to the single interaction between the two objects.
I'm not entirely sure I understand his second law and so let's move on.
The second answer is by Cleonis.
His second law is the standard second law. Cleonis claims that he's defining force and mass but he doesn't (well he does seem to imply that given a known force that mass is just a constant of proportionality in the last sentence of his third-to-last paragraph before the additional remarks -- but without a way of measuring force, it's not useful) -- so that's a bit disappointing.
Constantine then defines force as the time derivative of momentum.
Even if we leave force as a primitive we still need a method of measuring the mass of an object in any given system. His method of defining it in a system of two particles doesn't work if we have more than two particles so it has a very limited applicability.