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No, the Apple Watch series do not support eSIM, including Apple Watch 38mm, Apple Watch 42mm.
eSIM is a new technology that allows you to use mobile plans without inserting a physical SIM Card. The Apple Watch was designed to work with traditional nano-SIM cards only.
This new feature is only available in later models starting with the Apple Watch series 3.
Your current Apple Watch is not compatible with eSIM, but you can still use this new technology in 02 ways:
NOTE: Be aware that eSIM is not available everywhere or on every device. Your location or who you buy your phone from sometimes affects the availability of eSIM.