Locks have to be built out of something. What hardware primitives make synchronization possible?

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You cannot build mutual exclusion from ordinary loads and stores on modern hardware (reordering + interleaving defeat every classroom algorithm). CPUs therefore provide atomic read-modify-write instructions — test-and-set, fetch-and-add, and the workhorse compare-and-swap (CAS: 'if memory still holds A, replace with B, atomically'). Every mutex, semaphore, AtomicInteger and lock-free queue bottoms out in these, plus memory barriers to constrain reordering. Atomics work by the core claiming the cache line exclusively (via coherence) for the duration — which is also why contended atomics are expensive: cache-line ping-pong.