Is it really possible to obtain a AC signal (voltage/current) from a pure DC signal?
Yes and no.

In the real world, there's no such thing as "pure DC". Any real voltage source must be built and turned on, which is a transient event (AC). Real circuits also receive noise from background radiation and other sources. These things help keep oscillators from getting stuck in a metastable state.

In mathematical theory, "pure DC" is eternal -- it never begins or ends. A theoretical DC-powered oscillator must also be eternal. But if the DC voltage was always there, and the oscillation was always happening, I'm not sure it makes sense to say that the oscillation comes from the DC source. Can one eternal thing cause another?

As a practical matter, we ignore the philosophical questions. Yes, you can make a time-varying signal from a DC voltage supply. A simple example is a ring oscillator, which connects digital inverters in an unstable way:



There will usually be R-C filters on the inverter outputs to slow down and stabilize the oscillation.