Possible answer:

One physical quantity of the dimensions of length that can be formed out from c, G, and e²/4πε₀ is the Compton wavelength of an electron, λc, which is the characteristic distance scale associated with the wave-particle duality of matter and radiation.

The Compton wavelength of an electron can be derived by combining the principles of special relativity and quantum mechanics, and it depends only on fundamental constants that are universal and independent of any particular system or observer.

To derive λc, we can follow these steps:

1. Define the rest mass energy of an electron, mc², as the energy equivalent of its mass according to Einstein's famous formula, E = mc². This quantity has the dimensions of energy, but we can express it in terms of length by using Planck's relation between energy and frequency, E = hf, where h is Planck's constant and f is the frequency of the associated matter wave. Thus, mc² = hf, or f = mc²/h.

2. Apply de Broglie's hypothesis that matter particles have wave-like properties, such that their wavelength, λ, is related to their momentum, p, by λ = h/p. This relation implies that the momentum of a photon with the same frequency as the matter wave is also h/λ.

3. Use the principle of conservation of energy and momentum to consider a scenario where an electron collides with a photon of the same frequency in a head-on collision. In the rest frame of the electron, the photon has momentum p = mc, where c is the velocity of light, and the electron is at rest, so it has momentum p = 0. In the laboratory frame, the photon has momentum p' = mc, but the electron also has momentum p' = mev, where me is the relativistic mass of the electron and v is the velocity of the electron after the collision.

4. Apply the relativistic Doppler effect to relate the frequency of the scattered photon to the velocity of the electron, taking into account that the photon loses energy and momentum to the electron. This effect gives a formula for the wavelength shift, Δλ/λ = 1 - cosθ, where θ is the scattering angle between the incident and scattered photon.

5. Equate the initial and final energies and momenta of the electron and photon using the conservation laws, and solve for the velocity of the electron in terms of the scattering angle and the fundamental constants c, G, and e. This step involves some algebra and calculus, but the result is that v/c = (1 - cosθ)/[1 + (me/mc)(1 - cosθ)], where me/mc is a small dimensionless quantity that accounts for the relativistic mass increase of the electron.

6. Substitute this expression for v/c into the formula for λ in terms of p and the fundamental constants, and simplify the result to obtain the Compton wavelength of an electron, λc = h/mec, which is about 2.43 x 10^-12 meters or 0.0024 angstroms.

In summary, the Compton wavelength of an electron is a physical quantity of the dimensions of length that can be formed out from c, G, and e²/4πε₀, and it represents the scale at which the wave-like behavior of electrons becomes significant in their interactions with photons or