Consider the Fermi temperature of an ideal electron gas
$$T_{F} = \frac{\pi^2 \hbar}{2 m_e k_B} \left( \frac{3 n}{\pi} \right)^{2/3}$$
Or
$$T_F \sim \left(\frac{n}{10^{21} m^{-3}} \right)^{2/3} K$$
I.e. to make the Fermi temperature of the order of normal earthly temperatures of hundreds of Kelvin we need a density like $10^{24}$ electrons per meter cubed.
But this is not too difficult to reach, we know that one mole of matter, typically a small handful for normal solid elements, is $\sim 10^{23}$ atoms. We just need the electrons to be free, unconstrained within the atoms. We know of a case where electrons are almost freely flying around in a material, and that is metals. For instance in Magnesium, the density of conducting (free-ish) electrons is $\approx 8.6 \times 10^{28} m^{-3}$ so the electrons will be well degenerate even at room temperatures.
As a result, we can study the contribution of the degenerate electron gas to properties of metals. One historically very important result verified also experimentally is the fact that the heat capacity of most metals follows a law
$$C_V = K_1 T + K_2 T^3$$
where $K_1$ corresponds to the contribution of the degenerate electrons and $K_2$ to the "normal" oscillations of the grid (phonons) in the Debye model.
Similarly, one finds "hard" contributions to compressibility from the degenerate pressure of the electron gas. The pressure of the free non-relativistic ideal electron gas is
$$ P = \frac{\pi^2 \hbar}{5 m_e} \left( \frac{3}{\pi} \right)^{2/3} n^{5/3} $$
The compressibility is conventionally expressed using the isothermal bulk modulus $K_T$
$$K_T \equiv n \frac{\partial P}{\partial n}|_{T=const}$$
which gives us an estimate on the degenerate contribution as
$$K_T = \frac{\pi^2 \hbar}{3 m_e} \left( \frac{3}{\pi} \right)^{2/3} n^{5/3}$$
which is independent of temperature in the degenerate limit we are considering. This contribution to compressibility turns out to be in fact dominating to the compressibility of most metals.
You can contrast this with the compressibility of a classical ideal gas, where the same computation goes as
$$P = n k_B T \to K_T = n k_B T$$
and the compressibility scales linearly with temperature.
A pedagogical introduction to the topic and how you get beyond the free, noninteracting electrons is given in Chapters 6,7 and 9 of Kittel's Introduction to solid state physics, and a more advanced treatment is in the first half of the canonical Solid state physics by Ashcroft and Mermin.
