The numbers in notations like $|n\rangle$ are the analogues of indices in matrix notation. That is, $|0\rangle=e_0$, $|1\rangle=e_1$, etc., where $e_n$ is the vector which has a $1$ in the $n$th position and $0$ in the other entries. Unfortunately, this notation is unspecific about the dimension of the base space. For qubits in quantum computers, the dimension is $2$, so we only have $|0\rangle=e_0=(1,0)$ and $|1\rangle=e_1=(0,1)$. It is also common to have a countable infinity of basis vectors, so we get $|n\rangle$ for each $n\in\Bbb N$. In quantum mechanics one also deals with this notation for larger dimensional spaces; for example we may have $|x\rangle$ for each $x\in\Bbb R^3$ (the position basis), which is a vector space of uncountable dimension $|\Bbb R^3|=2^{\aleph_0}$.
In any case, these vectors are usually enumerating a basis of some kind, and the details beyond that depend on the context.
The notation $\langle 0|0\rangle$ is written in linear algebra notation as $e_0^Te_0$, which is a $1\times 1$ matrix whose value can be identified with the dot product $e_0\cdot e_0$. Provided that the vector is normalized, this will always be $1$. So a general answer is $\langle m|n\rangle=0$ if $m\ne n$, and $\langle n|n\rangle=1$, which expresses that the vectors $(|n\rangle)_{n\in\Bbb N}$ are an orthonormal basis for the space.
For some general rules, then, we have $|n\rangle=e_n$ and $\langle n|=e_n^T$ (or $e_n^\dagger$ in complex vector spaces), where we understand the first as a $d\times 1$ matrix so that the second is $1\times d$, where $d$ is the dimension of the space. Then the inner product is $\langle m|n\rangle=e_m^Te_n=e_m\cdot e_n$, and the outer product is $|n\rangle\langle m|=e_ne_m^T$, which is a $d\times d$ matrix with a single $1$ at the index $(n,m)$. Note that these notations are also used for arbitrary vectors; for example we might write $|\psi\rangle=v$ for some vector $v$, and then $\langle\psi|=v^T$, $\langle\psi|\psi\rangle=\|v\|^2$, and $|\psi\rangle\langle\psi|$ is the projection matrix in the direction of $v$.
Best Answer
The translation that Tyma Gaidash links to in the comments explains what these terms mean: they are the orbital positions of Venus, Earth, Jupiter, and Saturn at time $t$ years. So the value for a given planet is $360^\circ/P_\text{orb}$, where $P_\text{orb}$ is its orbital period in earth years.
The Royal Astronomical Society of Canada's Calgary Centre has this web page which gives the orbital periods of the planets in years to a very high precision (strangely, it gives the orbital period of Earth as $1.0000007$ years, which perhaps somebody can explain in the comments):
$$\begin{array}{c|c|c|} & P_\text{orb} & 360^\circ/P_\text{orb} \\ \hline \text{Venus} & 0.61517237 & 585.2018^\circ \\ \hline \text{Jupiter} & 11.8663142 & 30.3380^\circ \\ \hline \text{Saturn} & 29.47305083 & 12.2145^\circ \\ \hline \end{array}$$
This clearly shows that "585° dot 26" should be interpreted as $585.26^\circ$, and similarly for Jupiter and Saturn. The third column matches your quoted figures to better than one part in $2000$; if we interpreted the last two digits as arcminutes, the match would be no better than one part in $80$.
Updated to add: The Calgary Centre's Larry McNish was kind enough to reply to my query about that $1.0000007$: