Check if your tail lights work. The dashboard lights and the tail lights are (in most cases) commonly wired so you have at least feeble indication that you're invisible from behind at night.

Other than that: could be a faulty multi-pin connector where the wires plug onto your instrument cluster circuit board behind the dash. Could also easily be coincidence – crummy dash light rheostat in the headlight switch kills the dash lights, and faulty (woe is you) or ungrounded sender or dead gauge head kills the fuel gauge.

Does the engine temperature gauge work, or is it dead, too? Because if it is, that's usually the instrument cluster voltage limiter. Built into the gauge on the '63, not a separate plug-on unit, but that's easily reworked.