Running InfluxDB 2.0 and Telegraf using Docker allows developers to deploy these applications in a containerized environment, providing isolation, ease of deployment, and management. To do this, Docker must be installed on the local machine, and then the official images for Telegraf and InfluxDB 2.0 must be pulled from the Docker Hub. A new Docker network is created, and containers are run with specific arguments to connect them to the network and expose ports. Persistent storage is configured by mounting volumes to the container's configuration directories. The InfluxDB UI can be accessed at http://localhost:8086, and Telegraf configuration parameters must match InfluxDB setup parameters. Docker Compose can also be used to define and deploy multiple services, including InfluxDB, Telegraf, and an influxdb_cli service for setting up the InfluxDB instance.