Multi-step manipulation in dynamic environments remains challenging. Imitation learning (IL) is reactive but lacks compositional generalization, since monolithic policies do not decide which skill to reuse when scenes change. Classical task-and-motion planning (TAMP) offers compositionality, but its high planning latency prevents real-time failure recovery. We introduce SymSkill, a unified framework that jointly learns predicates, operators, and skills from unlabeled, unsegmented demonstrations, combining compositional generalization with real-time recovery. Offline, SymSkill learns symbolic abstractions and goal-oriented skills directly from demonstrations. Online, given a conjunction of learned predicates, it uses a symbolic planner to compose and reorder skills to achieve symbolic goals while recovering from failures at both the motion and symbolic levels in real time. Coupled with a compliant controller, SymSkill supports safe execution under human and environmental disturbances. In RoboCasa simulation, SymSkill executes 12 single-step tasks with 85% success and composes them into multi-step plans without additional data. On a real Franka robot, it learns from 5 minutes of play data and performs 12-step tasks from goal specifications. Code and additional analysis are available at https://symskill.github.io/ .