Is it possible to get the approximate XML file size from XSLT? Is it posible
to use some string functions to count the number of chars under document
element (though sounds terrible)?

Note that an XSLT processor will never see spaces in tags, for
example
 <foo bar="baz">
and
 <foo               bar="baz">
can't be distinguished. Other problems are entities and
character references:
 A
and
 &#x00000000000000000000000041;
is the same (ok, I may have exaggerated the leading zeroes a bit).