Context : The visual A component of the Gliese 586AB system is a double-lined spectroscopic binary consisting of two cool stars with the exceptional orbital eccentricity of 0.976 .
Such an extremely eccentric system may be important for our understanding of low-mass binary formation .

Aims : Precise stellar masses , ages , orbital elements , and rotational periods are a prerequisite for comparing stellar observations to angular-momentum evolution models .

Methods : We present a total of 598 high-resolution échelle spectra from our robotic facility STELLA from 2006–2012 which we used to compute orbital elements of unprecedented accuracy .
New Johnson VI photometry for the two visual components is also presented .

Results : Our double-lined orbital solution for the A system has average velocity residuals for a measure of unit weight of 41 m s ^ { -1 } for the G9V primary and 258 m s ^ { -1 } for the M0V secondary , better by a factor \approx 10 than the discovery orbit .
The orbit constrains the eccentricity to 0.97608 \pm 0.00004 and the orbital period to 889.8195 \pm 0.0003 d. The masses of the two components are 0.87 \pm 0.05 M _ { \odot } and 0.58 \pm 0.03 M _ { \odot } if the inclination is 55 \pm 1.5 ^ { \circ } as determined from adaptive-optics images , that is good to only 6 % due to the error of the inclination although the minimum masses reached a precision of 0.3 % .
The flux ratio Aa : Ab in the optical is between 30:1 in Johnson- B and 11:1 in I .
Radial velocities of the visual B-component ( K0-1V ) appear constant to within 130 m s ^ { -1 } over six years .
Sinusoidal modulations of T _ { eff } of Aa with an amplitude of \approx 55 K are seen with the orbital period .
Component Aa appears warmest at periastron and coolest at apastron , indicating atmospheric changes induced by the high orbital eccentricity .
No light variations larger than approximately 4 mmag are detected for A , while a photometric period of 8.5 \pm 0.2 d with an amplitude of 7 mmag is discovered for the active star B , which we interpret to be its rotation period .
We estimate an orbital period of \approx 50,000 yr for the AB system .
The most likely age of the AB system is \geq 2 Gyr , while the activity of the B component , if it were a single star , would imply 0.5 Gyr .
Both Aa and B are matched with single-star evolutionary tracks of their respective mass .

Conclusions :