Diode logic with an amplifier between each stage (https://en.wikipedia.org/wiki/Diode_logic) or DTL. Discrete surface mount diodes are still present on circuit boards and can be extracted for re-use. See the IBM 1401 for an example of a simple DTL computer.
Relay logic (https://en.wikichip.org/wiki/relay_computer). Building relays by hand using scrap material is practical, though difficult. Expect your gizmo to have the size, capability, and power requirements similar to the Zuse Z2 ( https://en.wikipedia.org/wiki/Z2_(computer) ).
* How would you build a transistor?
* How would you build thousands of transistors?
* How would you make them reliable?
* How would you power them?
* How would you implement memory?
Depending on what materials you have to hand, and your abilities to machine them, and your power sources, many alternatives present themselves. You could even consider analog computers.