Here is a first attempt, for others to improve on. For simplicity, I'll take $n=2k+1$ to be odd. We index the teams by integers modulo $n$.

We'll schedule $n$ rounds of $k$ games each. In the $j$-th round, the games (in order) are $$ \{ j+1, j-1 \},\ \{ j+2, j-2 \},\ \{ j+3, j-3 \},\ \ldots,\ \{ j+k, j-k \}$$ and team $j$ sits out.

The longest break between games is $2k$ games (team $j$ plays the first game in the $(j-1)$-st round, sits out the $j$-th round) and plays the first game in the $(j+1)$-st round). All the other breaks are either $k-1$ or $k+1$ games. So the ratio between longest break and shortest break is $\approx 2$.

I bet we can get that ratio down to $1+o(1)$. What solutions have other people found?

Here is a first attempt, for others to improve on. For simplicity, I'll take $n=2k+1$ to be odd. We index the teams by integers modulo $n$.

We'll schedule $n$ rounds of $k$ games each. In the $j$-th round, the games (in order) are $$ \{ j+1, j-1 \},\ \{ j+2, j-2 \},\ \{ j+3, j-3 \},\ \ldots,\ \{ j+k, j-k \}$$ and team $j$ sits out.

The longest break between games is $2k$ games (team $j$ plays the first game in the $(j-1)$-st round, sits out the $j$-th round and plays the first game in the $(j+1)$-st round). All the other breaks are either $k-1$ or $k+1$ games. So the ratio between longest break and shortest break is $\approx 2$.

I bet we can get that ratio down to $1+o(1)$. What solutions have other people found?