What kind of eel?

Many do burrow into the substrate. If you have an undergravel system, many will go down the tube and live out their lives underground.

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They are very shy. (I have only seem them scuba diving, never in a tank.)

Most eel species burrow but a few simply appropriate burrows, or as this site puts it:

“While some are pelagic, most are found living in small openings in coral reefs and rocks or burrowing in soft substrates.”

So, it rather depends on what type of eel you have. However, many of the burrowers come out of their burrows at night to hunt. They are also famous escape artists. We used to have difficulties with eels managing to get into the filter tubing and causing all sorts of problems for themselves as well as for the pump.

Interestingly enough, the spaghetti eels burrow, but unlike most species do so head first instead of tail first.