How many g's can astronauts withstand?

The question has two answers, depending on which phase of a flight is measured. Between liftoff and orbit there's a world of difference.

As soon as the engines shut off, the spacecraft falls freely around the Earth. The crew, the vehicle and any loose tool fall at the same speed, so nothing presses against a surface anymore. The onboard instruments only record minimal residual accelerations caused by air resistance, attitude control and crew movements. For the body, that's the real stressor: fluids shift upward, muscles and bones stop bearing load, and the balance organ stops receiving useful signals.

Only a few minutes per mission are uncomfortable. During ascent, acceleration increases as the rocket gets lighter by burning fuel; on return, the atmosphere slows the capsule down. In both cases, astronauts lie on their backs, so the forces act from chest to back. In that position, the circulatory system withstands the load much better than standing, because blood doesn't pool in the legs at the head's expense. Molded seats, bent legs and controlled breathing are part of the standard repertoire.

For preparation, centrifuges are used, where candidates are exposed in a controlled way to high accelerations, and parabolic flights: there, about 20 seconds of weightlessness alternate with pull-up phases of about 1.8 g, in which arms and head feel noticeably heavy. Some programs also include parachute jumps, because they train orientation under changing loads. After months in orbit, even normal Earth gravity is difficult: that's why returning crews are carried out of the capsule.