Hi, I recently bought an apartment, and the wiring has a 70 amp double pole main breaker.

I currently have 2 hots, 1 red, 1 black.

Do I have 70 amps per phase, or 70 amps total?

I believe it is wired with 3 or 4 gauge wire.

For example, if I run 50 amps on phase 1 and 50 amps on phase 2, will the 70 amp main breaker pop?

There is 70 amps per leg, not phase. The two wires, black and red deliver 70 amps at 240 volts, or 16,800 watts. There is also a white wire, or at least should be, which creates 120 volts between each the red and black to the white. You can actually have 70 amps per leg at 120 volts, which is 140 amps of 120 volt loads, or 16,800 watts.

A circuit should not be loaded more than 80% of it's rating if the load will operate 3 hours or more, considered continuous. Most electricians derate all wiring 80%, just as an extra safety measure.

50 amps on each leg should not trip a 70 amp breaker.