In his autobiography 'Things the Grandchildren Should Know', E reflects on the show at the Royal Albert Hall in London during the Eels with Strings tour in 2014.
He writes the following about 'My Beloved Monster' (p. 231, my emphasis):
I'm thinking about the irrational crush I had on the girl at the post office back in Virginia. I'm glad she's not here to hear me utter the embarrassing words I wrote about her.
Earlier on in his book (p. 109), E talks about the conception of the song:
One of the songs I recorded came about when my friend Jon Brion came over to my house one night. [..] He came over and suggested, as an exercise, that he would go upstairs for thirty minutes to write a song while I went downstairs for thirty minutes to write a song. [..] I went downstairs, picked up my paisley Telecaster guitar, plugged it into the tape recorder and started singing [..].
As for the girl the song is about (pp. 72-73):
I developed an irrational and embarrassing crush on a girl who worked at the post office. I only ever made small talk with her while buying stamps or sending mail at her counter, but I was always thinking about her. After months of this, I got the nerve to ask her out. She went out with me the day the space shuttle exploded and let me know that she was engaged. I went home and wrote a song called 'The Girl at the Post Office is Getting Married' and that was that.
Just to address the erroneous idea that the song is about E's sister Liz's suicide: this is impossible, as it took place shortly before Beautiful Freak was released (as pointed out in the comments by user Angst).
From page 119:
One night before the album's release, we played a show at the Alligator Lounge [..] I came home after the show and checked my phone messages. There was a message from my mom, who sounded really weird, saying to call her. [..]
[E.'s mother on the phone with him:] 'She took a bottle of pills and went into a coma ... and she ... she ... she ...'
There was a long pause. Then, 'she died.'
And as can be gleaned from the lyrics, the song is also quite unsuitable for the love for a sister:
My beloved monster is tough,
if she wants she will disrobe you.
But if you lay her down for a kiss,
her little heart it could explode.
This is one of E's most beloved songs, and he plays it (almost) every show, in a wide variety of ways.
On the Live at the Town Hall LP version from 2006, E included a new verse:
When she turns out the light every night,
she don't have a thing to fear.
Although the day didn't go quite right,
there's another chance near.
Something that shines through in his autobiography, is that E has the tendency to embellish the truth when he wants to get a certain idea across. From this verse it seems the subject of this song also has moved past the post office girl crush.