What does is really mean to say that a 3-body problem is not solvable? What does it really mean to say that a three-body problem (the Sun, the earth, and the moon) is not solvable? Why is it not possible to solve the differential equations on a computer with adequate initial conditions? What's the real issue here?
 A: Quoting from wikipedia

The three-body problem is a special case of the n-body problem. Unlike two-body problems, no general closed-form solution exists,[1] as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.

Of course, you can get a numerical solution for given initial conditions and masses. (Although, as Jon Custer points out, getting accurate numerical solutions is very difficult).
A: There has been always discussion about what is called solvable in the community.
For example: Take the pendulum with a constant force acting on it (and not applying the small angle approximation, otherwise the solution is a simple sine function), which is given by
$$  {\ddot {x}} = -\sin(x) \,.$$
With the small angle approximation
$$ {\ddot {x}} = -x\,,$$
you can solve this directly with the well-known function
$$ x = \sin(t) + \phi_0\,.$$
However, you can solve the full (non-linear) equations of motion as a perturbation series.
One could take the ansatz of writing $x$ as
$$ x = \sum_{\nu=0}^N \alpha_\nu \frac{t^\nu}{\nu!}$$
To lowest order in time $t$ we would then have
$$ \alpha_2+ \alpha_3 t +\mathcal{O}(t^2) = -\sin(\alpha_0+\alpha_1 t + \mathcal{O}(t^2) ) $$
which gives with the Taylor expansion of the sine
$$ \alpha_2 +\mathcal{O}(t)  = -\alpha_0+\mathcal{O}(t)  $$
You can then solve this equation order by order for each $\alpha_\nu$.
This can be done up to any order you want. Some people would say that this is not an analytical solution, but keep in mind, that depending on your philosophy the sine function is just a name for some perturbation series that has a particular structure. Just because the perturbation series that solves the above differential equation does not have a name, does not make the solution any less important.
Now addressing the question: You can solve the differential equation describing the three body problem numerically, but people usually mean that there exists no well known function that describe the solution in easy terms. There is no 'sine' function for the three body problem. Note that this is different from the two body problem, where we know the solution of $r(\phi)$ to be quite simple.
Additionally, the three body problem is chaotic (as was pointed out by Andrew), so any small change in the initial conditions will lead to large deviations for predictions far in the future. Numerically, chaotic systems are not easy problems, since you do not want unphysical choices (like the time step you choose) to influence your final result, but this happens often in simulations of these systems.
A: The three-body problem lacks a closed-form solution, which is a mathematical expression that uses a finite number of "standard" operations (addition, division, logarithm, etc.), usually expressed as a formula or equation. This means there is no equation for which you can plug in the initial positions, velocities, and masses, and solve for the exact positions and velocities at a later time. For most configurations of the three bodies, numerical methods are needed to iteratively compute the positions over time, although the chaotic behavior of the system means that even small numerical errors can propagate and result in large deviations over time.
Chaotic systems are very sensitive to their input parameters, so anything less than a perfect continuous estimation can result in wildly different results. Unlike a non-chaotic problem where a 1% error in velocity might result in a 1% error in position, a 1% error in velocity at some time step can result in arbitrarily large deviation at a later time. Even if your numerical approximation is very good, you may eventually find that your model predicts something entirely different from reality. You can run increasingly good approximations with decreasingly short time steps, but there is no numerical approach that is anything but an approximation, which simply may not be "good enough" in a chaotic system.
Of course, the three body problem does have a solution - it's whatever would actually happen when observing the three bodies in isolation. The physical reality is the solution to the problem. It's just that this solution cannot be expressed with common mathematical operations in "closed form".
See Why does the three-body problem have no solution?
A: It is Solvable...

Why is it not possible to solve the differential equations on a
computer with adequate initial conditions?

It is possible. But you will be hard-pressed to build a computer which can represent the "adequate initial conditions". We should say that a problem $P(x)$ is "solvable", if, for any input $x$, you give me a precise answer for $P$.
In the case of the sun-earth-moon system, that means that for arbitrary starting conditions and an arbitrary point in the future, you can tell me the positions of each body. For any system with precise dynamical equations, you can just plug in the initial conditions and iterate to the desired endpoint. What closed-form solutions give you is the ability to jump directly to the endpoint without performing any of the intermediate calculations. Obviously, this is very powerful. But it's more than that. It may be the only way to obtain a correct solution at all.
...In the Integers
The problem with a computer (all computers, including you) is that it can only compute with integers. IEEE 754 makes it look like you can compute with reals, but you are not. Floating-point numbers are just integers dressed up to look like reals, but actually doing a very poor job of it. You can perform exact symbolic operations on reals in the manner of Mathematica and friends, but the set of reals which can be handled in this manner is infinitely smaller than the full set of reals (due to the fact that all such reals must have a finite-length symbol attached to them, which just puts them in correspondence with the integers). Which means, the operations you can perform may quickly get exhausted while trying to obtain the answer you are looking for.
If your universe is quantized, and thus can be exactly described by integers, then, in principle, all 3-body problems should be exactly solvable. If time or space cannot be exactly represented by integers, then any computations you perform on them will contain errors. And any sequence of computations will accumulate growing errors.
Exponential Error
Some problems are sufficiently simple that the errors tend to cancel out, and one can feasibly obtain answers of arbitrary precision by performing computations of arbitrary precision (the precision of the result is proportional to the precision of the inputs). Simple harmonic motions, two-body problems, etc. fall into this category. When this happens, there is usually a closed-form solution which lets you jump straight to the desired endpoint.
When a physics problem is "not solvable", it generally means that despite an exactly precise dynamical description of the system (so far as we can tell, anyway), we cannot compute future states with arbitrary precision, because the accumulated error may grow without bound (i.e., "chaotic" systems). You can always increase the precision of the result by increasing the precision of the inputs and intermediate calculations; but if the error grows exponentially, then it quickly becomes infeasible to meaningfully improve the precision of the result by increasing the precision upstream of it. And this is what I mean by having trouble with "adequate initial conditions". If you want 64 bits of precision in your result, you might need something like 2^64 bits of precision in the initial conditions and intermediate calculations to achieve that. You can't build a computer big enough to get that answer.
That being said, some 3-body systems have "nice properties" which make their behavior not too chaotic. The bodies in the sun-moon-earth system are of masses and distances that we can compute useful ephemerides over most of the time periods of interest. Whereas, trying to determine which comets might collide with a planet in the next billion years might not be feasible at all, even given arbitrarily precise orbital elements.
