What was known about the properties of the nucleus (its shape, its density etc) and the nuclear forces before the Liquid drop model was proposed? I believe that some empirical knowledge must be out there already which motivated that the nuclei behave like a drop of liquid.
Sizes and masses were roughly known since Rutherford's 1911 experiments. Gamow references some more precise measurements from 1920s in Mass Defect Curve and Nuclear Constitution. For example, size measurements of Bieler from 1924 and Hardmeier from 1927 for light elements, and of Houtermans with Atkinson and with him from 1928-9 for heavy elements. The data on mass defects was taken from Aston's 1927 Bakerian Lecture on mass-spectrograph published in the Proceedings of Royal Society, which summarizes data accumulated since 1921.
Gamow also describes the interplay of attraction and repulsion forces known to exist between $\alpha$-particles in the nucleus as motivating his analogy to surface tension. His goal was to theoretically reproduce the experimental mass defect curve, but the motivation came from a theoretical analogy:
"In the present paper I shall attempt to treat the problem more closely, analysing from the theoretical point of view the experimental facts concerning the nuclear energy. To begin with we shall treat a nucleus built from a certain number of $\alpha$-particles only... The modern quantum theory of interaction between two complex particles gives a rather complicated expression for the mutual potential energy. We have here two kind of forces corresponding to symmetrical/antisymmetrical solutions of the wave equation. Both solutions show a strong repulsion at the distances compared with the dimension of particles. At greater distances the symmetrical solution gives an attraction decreasing exponentially with distance, the antisymmetrical one a repulsion of the same type...
Examining the behaviour of a collection of particles attracting one another with the forces very rapidly decreasing with distance (we neglect at first the Coulomb forces which are comparatively small at nuclear distances) we can introduce the well-known ideas made use of in the theory of capillarity".
What was known about the properties of the nucleus (its shape, its density etc) and the nuclear forces before the Liquid drop model was proposed?
In 1897 Thomson discovered the negatively charged electron. This prompted a revolution in thinking about the structure of the atom which before then had hardly been theorised. Thomsons discovery suggested that the neutral atom must contain a positively charged element. In 1903, Thomson proposed his Plum-Pudding Model - that the atom was a sphere of uniform positive electrification, with electrons scattered through it like plums in a pudding, giving rise to name.
In the following year, the Japanese physicist, Hantaro Nagaoka, rejected this model argung that charged elements must be impenetrable. In it's stead, he proposed a planetary model, in which a positively charged center is surrounded by a number of revolving electrons, in the manner of Saturn and its rings - prompting th name - The Saturnian Model. His model made two predictions:
a very massive atomic center (in analogy to a very massive planet)
electrons revolving around the nucleus, bound by electrostatic forces (in analogy to the rings revolving around Saturn, bound by gravitational forces).
Both of these predictions were successfully confirmed by Ernest Rutherford in his 1911 experiments and who mentions Nagaoka's model in his paper of that year - where the model of an atomic nucleus is mentioned.