I'm clearly missing something here, but I'm trying to grasp basics of how antennas work.
Relating it to standing waves on a string of length L, the lowest frequency possible is a wavelength of 2L, due to boundary conditions. This would lead me to assume the same for antennas: smallest size is wavelength/2. But I'm aware that antennas can be 1/4 wavelength, or even smaller. Why is that?
A $\lambda/2$ antenna has a standing wave with a node at both ends. A $\lambda/4$ antenna has a node and an anti-node. Anything shorter can't form standing waves at that frequency, which means that some of the energy that we inject into the antenna at one end will be reflected. This reflected energy (or better, the reflected power) is not available to the electromagnetic wave that the antenna sends out. In other words, short antennas are less efficient than properly sized ones.
They are actually much less efficient. The theory of "small radiators" and "small scattering bodies" is called "Rayleigh scattering". It is the same phenomenon that turns the sky blue. If an antenna of length $l$ is very short compared to the wavelength, then the emitted power is reduced by a factor proportional to $(\lambda/l)^4$, i.e. an antenna that is ten times shorter than a properly sized $\lambda/2$ dipole has just one ten thousands the efficiency (all other factors being equal, which means that in reality we can compensate somewhat for "slightly short" antennas). This has downsides for applications that require the transmission of long wavelengths, but it is technologically actually advantages because it limits the electromagnetic emissions from electronic circuits significantly. As long as we can make them small compared to the wavelengths of the frequencies they are operating at, they are not causing unwanted emissions.
