You can make a HEAD request:
$ curl -X HEAD -I http://USER:PASS@localhost:5984/db/doc_id
HTTP/1.1 200 OK
Server: CouchDB/1.1.1 (Erlang OTP/R14B03)
Etag: "1-c0b6a87a64fa1b1f63ee2aa7828a5390"
Date: Tue, 17 Jan 2012 21:32:43 GMT
Content-Type: text/plain;charset=utf-8
Content-Length: 740047
Cache-Control: must-revalidate
The Content-Length header contains the length in bytes of the document. This is very fast because you don't need to download the full document.
But there's a caveat: Content-Length is the number of bytes of the utf-8 version of the document (see the Content-Type header); String.length is the number of 16-bit utf-16 code units in a string.
i.e., they are counting different things, bytes versus code units, of different encodings of the document, utf-8 versus utf-16.