Why is the predictability of the solar system in the Lyapunov timescale limited to 5 million years? Is this due to a mathematical problem that is not solved? Or could this be due to our current amount of information regarding mass and other such factors in our system?
 A: The tags Qmechanic added to the post already answer the question: the solar system is unpredictable in the long term because of the chaotic behavior of the many-body problem (where $\text{many}\ge3$).
In a chaotic system - and most natural systems are chaotic - any uncertainty, however small, is blown up eventually since it grows exponentially: thus, since uncertainties are unavoidable in practice (and even in principle), its predictability is severely limited. Do check Wikipedia's entries on Chaos Theory and on the Stability of the Solar System - the latter of which sums it up:

[the planets] weak gravitational effects on one another can add up in unpredictable ways. For this reason (among others) the Solar System is chaotic in the technical sense of mathematical chaos theory, and even the most precise long-term models for the orbital motion of the Solar System are not valid over more than a few tens of millions of years.

The specific Lyapunov time of 5 million years for the solar system, however, is rather a lower bound, that is, its orbits can be predicted for at least this period, with the exact value of the Lyapunov time being unknown at the moment (according to this 2007 paper [free version]).
Related questions:

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*How does chaotic situation arise in planetary motion in solar system?

*Earth's orbit: chaotic but stable

*If the solar system is a sensitive chaotic system, can gravitational waves make orbits unpredictable?
