How can one get the eccentricity of the orbit of the Sun around center of the Milky Way? Can it be measured?


Short answer, no.
The Sun's orbit is non-Keplarian; there are many perturbations and a general unevenness in the motion of the Sun around the Galactic centre. This is a result of non-uniform mass distributions, the galaxy not being a point mass, and the impact the relative motions of neighbour stars has on measuring. Thus, giving a particular eccentricity for the Sun is almost meaningless. For instance, it fluctuates up and down roughly $2.7$ times per orbit and it passes through high density regions which cause major perturbations. This creates instability in any average eccentricity.

Long answer, it is not impossible.
In theory, we could measure it. However, we have two rest frames; local and standard rest. The local rest frame refers to how we can take the average motion of stars within (say) $100~pc$ and use this average to compute our approximate orbital properties. The standard rest frame refers to us using Oort constants/properties and similar things in order to determine our more specific motion around the galaxy based on accelerational perturbations, etc. Both frames have their own advantages and both give slightly different values for our currently computed orbital characteristics. The problem lies with determining the relative weights each might contribute to an eccentricity value.

While the motion of the Sun may be non-Keplarian, we do know that the circular velocity is around $230{km\over s}$ and the peculiar velocity is on order $15{km\over s}$. This leads many astronomers to say that while measuring the eccentricity would be very hard and calculating it would be near impossible, they can say that it is most likely on the order of a few percent. Definitely less than $10\%$, but a value in the range of $e=0.02-0.08$ would be the most likely.

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