Short answer: no.
Long answer: there are ways to achieve this, but you'll have to sacrifice a lot of other fairly essential highlighting features.
There is certainly no way to do it with a simple option. The problem that you may come across is that Vim doesn't allow overlapping regions, so if you have rainbow.vim installed or anything else that makes a region in the area of your block, it's lost. It's also difficult (although I'd welcome any corrections) to have multiple highlighted groups, so one that sets the background colour and another that sets the foreground. This is very limiting as you'll see.
However, if fancy playing around, read on.
I'm making the assumption here that you're using C code in the same coding style as me, but this could easily be changed...
Here's a simple function that should help show you what's involved:
function! HighlightBlock()
" Search backwards and forwards for an opening and closing brace
" at the start of the line (change this according to your coding
" style or how you define a block).
let startmatch = search('^{', 'bcnW')
let endmatch = search('^}', 'cnW')
" Search in the other direction for each to catch the situation
" where we're in between blocks.
let checkstart = search('^{', 'cnW')
let checkend = search('^}', 'bcnW')
" Clear BlockRegion if it exists already (requires Vim 7 probably)
try
syn clear BlockRegion
catch
endtry
" If we're not in a block, give up
if ((startmatch < checkstart) && (endmatch > checkstart))
\ || ((startmatch < checkend) && (endmatch > checkend))
\ || (startmatch == 0)
\ || (endmatch == 0)
return
endif
" Create a new syntax region called "BlockRegion" that starts
" on the specific lines found above (see :help \%l for more
" information).
exec 'syn region BlockRegion'
\ 'start=' . '"\%' . startmatch . 'l"'
\ 'end=' . '"\%' . (endmatch+1) . 'l"'
" Add "contains=ALL" onto the end for a different way of
" highlighting, but it's not much better...
" Define the colours - not an ideal place to do this,
" but good for an example
if &background == 'light'
hi default BlockRegion guibg='#AAAAAA'
else
hi default BlockRegion guibg='#333333'
endif
endfunction
To use the function, source it from somewhere and then create an autocmd to call it whenever something changes, e.g.
au CursorMoved *.c call HighlightBlock()
See the following for some autocommands you may want to consider:
:help CursorHold
:help CursorHoldI
:help CursorMoved
:help CursorMovedI