There is no such universal constant $\epsilon > 0$. Work with the complex Hilbert space $L^2[0,1]$ (which of course is also a real Hilbert space). Fix $n \in \mathbb{N}$.

Let $\Gamma_0$ be the set of continuous piecewise linear increasing bijections from $[0,1]$ to itself. [1] It is a group with composition as product. [2] It acts by isometries of $L^2[0,1]$ by the map $f \mapsto \sqrt{\phi'}\cdot (f\circ\phi)$ for $f \in L^2[0,1]$ and $\phi \in \Gamma_0$. Also let $\Gamma_1 \subset L^\infty[0,1]$ consist of the measurable functions from $[0,1]$ to $\mathbb{T}_n = \{e^{2\pi i k/n}: 0 \leq k < n\}$, identifying functions which differ on a null set. This is a group under pointwise product and [3] it acts isometrically by multiplication on $L^2[0,1]$. Let $\Gamma$ be the group of isometries of $L^2[0,1]$ generated by $\Gamma_0$ and $\Gamma_1$ under these actions. ([4] This is a semidirect product of $\Gamma_0$ and $\Gamma_1$.)

[5] The $\Gamma_0$ action takes the unit vector $1_{[0,1]}$ to any piecewise constant strictly positive unit vector $f \in L^2[0,1]$. (If $f$ takes the value $c$ on an interval $I$, let $\phi$ have slope $c^2$ on this interval.) [6] These functions are dense in the positive part of the unit sphere. [7] Applying the action of $\Gamma_1$ then gets us to arbitrarily close to any unit vector in $L^2[0,1]$ whose argument lies almost everywhere in $\mathbb{T}_n$. [8] It follows that the distance from $1_{[0,1]}$ to any other orbit is at most $\alpha = |1 - e^{\pi i/n}|$ ($\approx \frac{\pi}{n}$ for large $n$). [9] It follows straightforwardly that the same is true for any positive unit vector in place of $1_{[0,1]}$, and then [10] that the distance between any two orbits is at most $2\alpha$.

[11] On the other hand, the distance from the orbit of $1_{[0,1]}$ to the vector $e^{\pi i/n}\cdot 1_{[0,1]}$ is at least the distance from $(1,0) \in \mathbb{R}^2$ to the line through the origin of slope $\frac{\pi}{n}$ (again approximately $\frac{\pi}{n}$ for large $n$), so this orbit is not dense and since the action is isometric no orbit is dense.

Edit: maybe people want more details.

[1] The composition of two continuous functions is continuous, of two PL functions is PL, of two increasing functions is increasing, of two bijections is a bijection. The inverse of a continuous PL increasing bijection is a continuous PL increasing bijection.

[2] $\|\sqrt{\phi'}\cdot (f\circ \phi)\|_2^2 = \int_0^1 \phi'|f\circ\phi|^2\, dt = \int_0^1 |f|^2\, dt = \|f\|_2^2$.

[3] If $h \in \Gamma_1$ then $\|hf\|_2^2 = \int_0^1 |hf|^2\, dt = \int_0^1 |f|^2\, dt = \|f\|_2^2$ since $|h| = 1$ a.e.

[4] This isn't needed, but anyway if $\phi \in \Gamma_0$ and $h \in \Gamma_1$ then $\sqrt{\phi'}\cdot (hf\circ \phi) = (h\circ \phi)\cdot \sqrt{\phi'}\cdot(f\circ\phi)$.

[5] Let $f$ be a piecewise constant strictly positive unit vector in $L^2[0,1]$. Then $f = a_0\cdot 1_{[0,t_1)} + \cdots + a_k\cdot 1_{[t_k,1)}$ a.e. for some $0 < t_1 < \cdots < t_k < 1$ and some $a_0, \ldots, a_k > 0$. The unit norm condition means that $\sum_{i=1}^k a_i^2(t_{i+1} - t_i) = 1$. Now define $\phi: [0,1] \to \mathbb{R}$ so that $\phi(0) = 0$, $\phi$ is continuous, and $\phi$ is linear with slope $a_i^2$ on $[t_{i-1},t_i]$. The unit norm condition just detailed shows that $\phi(1) = 1$, i.e., $\phi$ is a continuous PL increasing bijection. We have $\sqrt{\phi'}\cdot (1_{[0,1]}\circ \phi) = \sqrt{\phi'}$, which takes the value $a_i$ constantly on $(t_{i-1},t_i)$. So $1_{[0,1]}$ is taken to $f$.

[6] First, positive piecewise constant functions can uniformly approximate any continuous function on $[0,1]$, and since the positive continuous functions are dense for the $L^2$ norm in the positive part of $L^2[0,1]$, this shows that positive piecewise constant functions are dense in the positive part of $L^2[0,1]$. Given a positive $f \in L^2[0,1]$ with unit norm, find a sequence $(f_k)$ of positive piecewise constant functions with $f_k \to f$ in $L^2[0,1]$. Then $\|f_k\|_2 \to 1$ so $\frac{1}{\|f_k\|_2}f_k \to f$. Thus, any positive unit vector is approximated by positive piecewise constant unit vectors.

[7] Since multiplying by $h \in \Gamma_1$ is an isometry, if $f = h|f| \in L^2[0,1]$ is a unit vector whose argument $h$ lies in $\mathbb{T}_n$ a.e. and $g$ is a positive piecewise constant unit vector which is close to $|f|$, then $hg$ will be equally close to $f$.

[8] Given any unit vector $f = h|f| \in L^2[0,1]$, we can find $\tilde{h} \in \Gamma_1$ such that $|h(t) - \tilde{h}(t)| \leq \alpha$ a.e. As we just saw that the orbit of $1_{[0,1]}$ comes arbitrarily close to $\tilde{h}|f|$, it follows that the distance from $f$ to this orbit is at most $\|f - \tilde{h}|f|\|_2 = \|(h - \tilde{h})|f|\|_2 \leq \alpha \|f\|_2 = \alpha$.

[9] We saw already that any positive unit vector $f$ is approximated by elements in the orbit of $1_{[0,1]}$. Since the action is isometric, this means that $1_{[0,1]}$ is approximated by elements in the orbit of $f$. Again by isometric action, since we can take $1_{[0,1]}$ to within $\alpha'$ of any unit vector, for any $\alpha' > \alpha$, the same is then true of $f$.

[10] Any two unit vectors lie within $\alpha'$ of the orbit of $1_{[0,1]}$, for any $\alpha' > \alpha$. So (by isometric action again) one lies within $2\alpha'$ of the orbit of the other.

[11] The argument of any vector $f$ in the orbit of $1_{[0,1]}$ lies pointwise a.e. in $\mathbb{T}_n$. So $|f(t) - e^{\pi i/n}| \geq \beta$ pointwise, where $\beta$ is the distance from $(1,0) \in \mathbb{R}^2$ to the line through the origin of slope $\frac{\pi}{n}$ (= the distance from $e^{\pi i/n} \in \mathbb{C}$ to the union of the lines through the origin of slopes $\frac{2k\pi}{n}$). Thus $\|f - e^{\pi i/n}\cdot 1_{[0,1]}\|_2 \geq \beta$.

for every $\Gamma$ and every action, we have the given inequality. Also note that the statement "the orbit $X$ is at distance at least $\varepsilon$ from some other orbit" is equivalent to the statement "there exists a point $v$ (in the 1-sphere) at distance $\ge\varepsilon$ of $X$". This is true for arbitrary isometric actions on arbitrary metric spaces. $\endgroup$ – YCor Sep 22 '18 at 17:31