To solve problems in 3D space rather than merely describe images, multimodal large language models (MLLMs) need 3D spatiotemporal reasoning: reconstructing 3D geometry from occluded 2D views, anticipating physical dynamics, and executing precise, closed-loop actions. Prior studies test passive VQA on static images or symbolic planning, requiring no execution or supplying privileged coordinates while ignoring collision, statics, and timing. We introduce SpaceAct, an interactive benchmark evaluating 3D spatiotemporal reasoning via closed-loop physical action and active perception, with SAPA, a Spatiotemporal Active-Probing Agent framework. Built on interlocking bricks and mechanical puzzles, it has 108 verified instances across 36 task families in six categories, without privileged coordinates, and diagnostic modes (+plan, +state, no-camera, blind, L7 levers). In brick-bridging or aperture-threading, models rotate the camera to resolve occlusion, read in-scene rulers, output 6-DoF poses under swept-collision and statics checks, and self-correct via split-screen rollback feedback (dynamic tasks freeze time during deliberation and give five strobe frames). Across 10 frontier MLLMs on the 44 hardest instances, scores span 4.8-79.1 (seven solve at most 27.3% of 6-DoF tasks); for the top three, a reference plan (+plan) closes 44% of the gap while leaving collisions at 5.2 per episode, whereas adding exact 3D state (+state) closes 90% and cuts collisions to 0.5; for four others, +plan lifts the mean score from 10.9 to 35.3 and +state to 81.8; and 46% of failures are collision or observation loops. SpaceAct and SAPA advance evaluation to closed-loop physical interaction, showing that even with a valid plan, MLLMs remain bottlenecked by 3D visual grounding and action refinement, while offering a testbed for active multi-view exploration and explicit 3D grounding.