Unanticipated runtime errors, lacking predefined handlers, can abruptly terminate execution and lead to severe consequences, such as data loss or system crashes. Despite extensive efforts to identify potential errors during the development phase, such unanticipated errors remain a challenge to to be entirely eliminated, making the runtime mitigation measurements still indispensable to minimize their impact. Automated self-healing techniques, such as reusing existing handlers, have been investigated to reduce the loss coming through with the execution termination. However, the usability of existing methods is retained by their predefined heuristic rules and they fail to handle diverse runtime errors adaptively. Recently, the advent of Large Language Models (LLMs) has opened new avenues for addressing this problem. Inspired by their remarkable capabilities in understanding and generating code, we propose to deal with the runtime errors in a real-time manner using LLMs. Specifically, we propose Healer, the first LLM-assisted self-healing framework for handling runtime errors. When an unhandled runtime error occurs, Healer will be activated to generate a piece of error-handling code with the help of its internal LLM and the code will be executed inside the runtime environment owned by the framework to obtain a rectified program state from which the program should continue its execution. Our exploratory study evaluates the performance of Healer using four different code benchmarks and three state-of-the-art LLMs, GPT-3.5, GPT-4, and CodeQwen-7B. Results show that, without the need for any fine-tuning, GPT-4 can successfully help programs recover from 72.8% of runtime errors, highlighting the potential of LLMs in handling runtime errors.