Let's investigate the range of $x_1\le x_2 \le \cdots \le x_n$ given that their arithmetic mean (AM) is a small multiple $1+\delta$ of their geometric mean (GM) (with $\delta \ge 0$). In the question, $\delta\approx 0.001$ but we don't know $n$.
Since the ratio of these means does not change when the units of measurement are changed, pick a unit for which the GM is $1$. Thus, we seek to maximize $x_n$ subject to the constraint that $x_1+x_2+\cdots+x_n = n(1+\delta)$ and $x_1\cdot x_2\cdots x_n = 1$.
This will be done by making $x_1=x_2=\cdots=x_{n-1}=x$, say, and $x_n=z \ge x$. Thus
$$n(1+\delta) = x_1 + \cdots + x_n = (n-1)x + z$$
and
$$1 = x_1\cdot x_2 \cdots x_n = x^{n-1}z.$$
The solution $x$ is a root between $0$ and $1$ of
$$(1-n)x^n + n(1+\delta)x^{n-1} - 1.$$
It is easily found iteratively. Here are the graphs of the optimal $x$ and $z$ as a function of $\delta$ for $n=6, 20, 50, 150$, left to right:

As soon as $n$ reaches any appreciable size, even a tiny ratio of $1.001$ is consistent with one large outlying $x_n$ (the upper red curves) and a group of tightly clustered $x_i$ (the lower blue curves).
At the other extreme, suppose $n=2k$ is even (for simplicity). The minimum range is achieved when half the $x_i$ equal one value $x \le 1$ and the other half equal another value $z \ge 1$. Now the solution (which is easily checked) is
$$x^k = 1+\delta \pm \sqrt{\delta^2 + 2\delta}.$$
For tiny $\delta$, we may ignore the $\delta^2$ as an approximation and also approximate the $k^\text{th}$ root to first order, giving
$$x \approx 1 + \frac{\delta-\sqrt{2\delta}}{k};\ z \approx 1 + \frac{\delta+\sqrt{2\delta}}{k}.$$
The range is approximately $\sqrt{32\delta}/n$.
In this manner we have obtained upper and lower bounds on the possible range of the data. We have learned that they depend heavily on the amount of data $n$. The upper bound shows the range can be appreciable even for tiny $\delta$, thereby improving our sense of just how close to each other the data points really need to be--and placing a lower limit on their range, too.
Similar analyses, just as easily carried out, can inform you--quantitatively--of how tightly clustered the $x_i$ might be in terms of any other measure of spread, such as their variance or coefficient of variation.
x=c(-5,-5,1,2,3,10); prod(x)^(1/length(x))
$\:\quad$[1] 3.383363
(while the arithmetic mean is 1) $\endgroup$ – Glen_b Jun 28 '16 at 23:50