What you ask is a bad idea. For a while some external drives were sold with USB connections that did NOT need their own power supplies, because the standard USB port has power available - limited, but some. Especially as hard drives got larger, some appeared with a branched cable with TWO USB connectors on the end, and you had to plug in both to different sockets so that the drive could get enough power from two outlets - one was not enough. Some small external drives - the ones based on 2½" drives and sold for use with laptops -still use this dual connection system. But almost nobody makes cases with 3½" HDD's inside without their own power supply - you simply can't draw enough power from one (or even two) USB connectors.
The eSATA standard has NO power supplied in the data cable. On internal SATA the data and power supply cables are separate. eSATA does the same. So you could not attach an external hard drive by eSATA only - it would get no power. I suppose you could design a unit that communicates by eSATA but obtains power by two USB connections, but why would you? Firewire systems, on the other hand, DO have some power available in the full-size connector, but not in the small one used for self-powered peripherals. (That's similar to USB, which has a smaller no-power-available version of connector.)